Republican senators were split Sunday about whether President Trump’s former personal attorney, Michael Cohen, should return to Capitol Hill to testify after he claimed the president knew of his son’s meeting in Trump Tower with a Russian ahead of the 2016 election.

Mr. Cohen claimed last week Mr. Trump knew of the meeting between a Russian lawyer linked to the Kremlin and his son Donald Trump Jr. ahead of it taking place in the summer of 2016. The lawyer reportedly promised damaging information about Mr. Trump’s opponent Hillary Clinton.

The new revelation by Mr. Cohen, though, raises questions about whether Mr. Trump’s son committed perjury when he told lawmakers last year that his father had no prior knowledge of the meeting.

Mr. Trump and the White House also have denied knowing about the meeting ahead of time.

Sen. Lindsey Graham, South Carolina Republican, said Sunday that Mr. Cohen already has testified before Congress, but the information about Mr. Trump’s knowledge of the Trump Tower meeting was new to lawmakers.

“You need to come to either the Judiciary Committee, the intel committee or both,” Mr. Graham said during an appearance on Fox News, directing his statement to Mr. Cohen.

When Sen. Rob Portman, Ohio Republican, was pressed on the issue during an interview Sunday with NBC’s “Meet the Press,” he wasn’t eager to call Mr. Cohen back before Congress.

“I think the Mueller investigation is the place that this should be cleared up,” he said, referencing special counsel Robert Mueller’s probe into whether the Trump campaign colluded with Russians.

And Sen. James Lankford, a member of the Senate Select Committee on Intelligence, told ABC’s “This Week” that he also thinks Mr. Mueller’s probe is the right place for Mr. Cohen’s claim to be reviewed.

“That’s something that is going to be more into the lane of the special counsel than it will for the intelligence committee. We are working through the policy issues and Russian interference,” the Oklahoma Republican said Sunday.

Mr. Cohen’s new allegation comes after his attorney leaked a tape recording to CNN last week, which revealed then-candidate Donald Trump and Mr. Cohen discussing hush money for a former Playboy model who claimed she had an affair with the president.

The model ended up selling her story exclusively to the National Enquirer for $150,000, but the outlet never ran her story.

The president’s attorney Rudy Giuliani said Saturday he has warned Mr. Cohen to keep quiet, saying that Mr. Cohen is violating attorney-client privilege by releasing the tape.

But attorney Lanny Davis, who is representing Mr. Cohen, told reporters that Mr. Giuliani “expressly waived attorney-client privilege last week” by talking about the recording.

According to Mr. Giuliani, Mr. Cohen first said he was present in meeting where the president and several of his aides discussed the meeting between Mr. Trump Jr. and the Russian lawyer two days after the meeting, but recently Mr. Cohen has claimed he was present for a conversation with the president the day of the meeting before it took place.

“There was no such meeting in advance,” Mr. Giuliani told “Fox News Sunday.” “It’s just flat out untrue.”

Mr. Cohen reportedly is under investigation after the FBI raided his office in April and took business records and files from his hotel room in New York and his office.

Mr. Graham said he’s suspicious of Mr. Cohen’s new claims since he is currently in the hot seat, suggesting the leaking of the tape was a “media strategy” to get Mr. Mueller’s attention.

“The one thing about Michael Cohen is I’ve never seen a lawyer behave this way,” Mr. Graham said Sunday on Fox News.

The president retweeted one of Mr. Cohen’s tweets posted last July, in which he praised Mr. Trump Jr. for being “transparent.”

“Do you think the Fake News Media will ever report on this tweet from Michael?” the president tweeted Sunday, suggesting Mr. Cohen’s past tweet contradicts his new accusation.