"It's a big deal, and we want to make sure sensitive information is handled properly," a European diplomat told CNN, adding that the country represented by the diplomat would not cut communications with the US entirely but could potentially reassess what information is transmitted at the political level.
The diplomat -- who spoke to CNN on the condition that neither their name nor country were identified -- said allied agencies could explore ways to share specific information through channels used only by military and intelligence officials.
Another European diplomat told CNN that Europeans are expecting cooperative exchanges with the US on this matter and to avoid US unilateral measures.
An EU-US meeting on airline security and electronics devices Wednesday afternoon in Brussels will allow US deputy secretary Elaine Duke to share information with EU allies, the diplomat told CNN.
"It is somewhat surprising, though, that intel was shared on this issue, with Russian Minister (Sergey) Lavrov, before being shared with ministers of allied countries," the diplomat said.
But Defense Secretary James Mattis said that US allies had not expressed concerns to him about the possibility that sensitive intelligence may have been compromised, noting that he spoke to two NATO allies on Tuesday and "it never came up."
A third diplomatic source representing a country that shares intelligence with both Russia and the US told CNN that since Trump's reported revelations seemed to be a mistake, there is less concern than if the information was deliberately shared by the CIA to another country.
"He made a mistake ... It likely won't happen again," the diplomatic source said.
Ultimately, Trump's disclosure of sensitive information is unlikely to prevent future sharing of intelligence between the US and the source's country because cooperation is done for "mutual benefit," the source added.
Former defense secretary and CIA Director Leon Panetta told CNN's Chris Cuomo Tuesday that the fallout of breaching the trust of key allies by disclosing information could be significant.
"You don't just get intelligence out of thin air. You get intelligence because we deploy spies, because we deploy people who are willing to put their lives on the line and because we work with other intelligence agencies around the world that help provide that kind of information," he said on "New Day." "But it is done on the basis of competence and trust."
According to Panetta, the ally who provided the information that was reportedly relayed to the Russians could cut off any type of intelligence provided to the US in the future.
'We're not going to cut the cord'
But despite claims that Trump relayed information that could pose serious national security challenges for the US and its allies, the first European diplomat said officials are waiting to see how the situation plays out before making any decisions regarding the level of intelligence sharing with the US.
While there have been some early conversations about new ways to protect shared information, the diplomat emphasized that the professionalism and the scope of work by US intelligence officials make it essential to maintain close ties.
Like many US allies, the diplomat represents a country whose intelligence agencies work so closely with the US on issues like Syria that it would be very hard to stop that cooperation altogether.
Although pulling back "is something we could envision," the diplomat said, "we're not going to cut the cord."
The US is part of the so-called "Five Eyes" agreement (along with Britain, Canada, Australia and New Zealand), which calls for open sharing among member nations of a broad range of intelligence.
The US also maintains open intelligence-sharing relationships with allies like France, Germany and Japan. In the Middle East, the US formally and informally shares information with several countries in the fight against ISIS and other terror groups including Jordan, Egypt, Saudi Arabia, the United Arab Emirates and Turkey, among others.
Serious implications of Trump disclosure
Indications that allies are discussing the possibility of reconsidering their information-sharing relationships with the US comes in the wake of Monday's bombshell Washington Post report
, major details of which were confirmed by CNN, that Trump had shared with the Russians information provided by a foreign partner agency.
The information that "endangered cooperation from an ally that has access to the inner workings of the Islamic State," according to the Post, was reportedly so sensitive that details were withheld from other international partners and restricted even within the US government.
Trump's national security adviser, H.R. McMaster, said Tuesday that he stands by his previous statement on The Washington Post report -- he said the story "as reported is false" -- adding, "The premise of that article is false that in any way the President had a conversation that was inappropriate or that resulted in any kind of lapse in national security."
McMaster said he has not spoken to foreign nations regarding the reports and intelligence sharing.
Israeli intelligence was a source for some of the information ISIS bomb-making capabilities that the President reportedly discussed with Russian diplomats, according to US and diplomatic officials.
Israeli's ambassador to the United States, Ron Dermer, would not comment on the intelligence, but expressed confidence in the relationship with the US.
The potential consequences of disclosing such sensitive information could hardly be more serious, former CIA case officer Bob Baer told CNN's Erin Burnett on Monday.
"The President, by revealing this to the Russians, has lost control of this information. It's going to go to the Syrians, It's going to go to the Iranians -- Russian allies," Baer said.
"The ability to protect that source whoever he is, wherever he is has been seriously undermined. ... If a CIA officer had revealed this information to the Russians, he would be fired instantly."