For many Democrats, Joe Manchin has become an unshakeable problem. The centrist senator is at odds with other Democrats on everything from filibuster reform to climate policy, and he recently announced his opposition to the Build Back Better Act, the lynchpin of Joe Biden’s legislative agenda.

But Republicans think Manchin now represents an opportunity to boost their numbers.

As Democrats have leveled fierce criticism at the West Virginia senator in the past few days, Republicans have resurrected their campaign to recruit him to their party.

The stakes of this charm offensive could not be higher. With the Senate split 50-50, Manchin’s party change would give Republicans the majority. If Republicans take control of the Senate, they would have the ability to block Biden’s nominees and quash Democratic bills.

Speaking to the New York Times on Tuesday, the Senate minority leader, Mitch McConnell, reiterated his invitiation to Manchin to join the Republican caucus. “Obviously we would love to have him on our team,” McConnell said. “I think he’d be more comfortable.”

The Republican senator John Cornyn said he also texted Manchin on Tuesday to tell him: “Joe, if they don’t want you we do.”

Cornyn told the NBC affiliate KXAN that he had not heard back from Manchin, but he said a change in Senate control would be “the greatest Christmas gift I can think of”.

Manchin has not given any indication that he is seriously considering switching parties. In a Monday interview with West Virginia Radio, Manchin said he believed there was still room in the Democratic party for someone with his views.

“I would like to hope that there are still Democrats that feel like I do,” Manchin said. “I’m socially – I’m fiscally responsible and socially compassionate.” He added: “Now, if there’s no Democrats like that, then they’ll have to push me wherever they want me.”

Manchin has been even more pointed in the past when asked about his party identity. After a report emerged in October that he was seriously considering leaving the Democratic party, he dismissed the news as “bullshit”.

But he acknowledged he had previously offered to change his party affiliation to “independent” if his views ever became an “embarrassment” for Biden or other Senate Democrats.

“I said, me being a moderate centrist Democrat — if that causes you a problem, let me know and I’d switch to be independent,” Manchin said in October.

At the time, none of Manchin’s Democratic colleagues took him up on the offer, although some may now be tempted to do so. When Manchin announced he would oppose Build Back Better, after he had already demanded major changes to the spending package to limit its size and scope, some congressional Democrats sounded ready to abandon their colleague.

“It’s unfortunate that it seems we can’t trust Senator Manchin’s word,” Pramila Jayapal, the chair of the Congressional Progressive Caucus, said on Monday. “We’re not going to wait for one man to decide on one day that he’s with us, and on the other day that he’s not.”

For McConnell, that is an opening to try to wrest back control of the Senate.

“Why in the world would they want to call him a liar and try to hotbox him and embarrass him?” McConnell told the Times. “I think the message is, ‘We don’t want you around.’ Obviously that is up to Joe Manchin, but he is clearly not welcome on that side of the aisle.”