Sen. John CornynJohn CornynLawmakers feel pressure on guns Kasich’s campaign website tones down gun language after Florida shooting Murphy: Trump’s support for background check bill shows gun politics ‘shifting rapidly’ MORE (R-Texas) said Senate Republicans would "welcome" any Democrats who wish to switch parties and caucus with the GOP.

Cornyn, the head of Senate Republicans' campaign efforts, floated the possibility that the GOP might target Sen. Joe Lieberman (Conn.), an Independent who caucuses with Democrats, or another Democratic senator if Republicans come close to winning a majority but fall short.

"I think he votes like a Republican on those areas, and we would certainly welcome him or any other Democrat who wants to switch sides of the aisle and caucus with us," Cornyn said of Lieberman during an Election Day appearance on Fox News.

Republicans are optimistic that they will make major gains in the Senate in this year's mid-term election. They need a net gain of 10 seats to win a Senate majority, but Cornyn has said that such a lofty goal probably isn't realistic. Projection services have pegged Republican gains along the lines of six to eight seats.

Those sorts of gains would leave the GOP a few votes shy of the 51 seats they need to have a majority in the Senate (if the chamber is split, Vice President Biden, a Democrat, would break the tie). Republicans would need to pick off a few senators — Lieberman and maybe one or two others — to get a majority.

"I think after this election, we're going to have to look and see who's left, because I think there are going to be a lot of people who aren't coming back, and find out who we can work with," Cornyn said.

The GOP might look to other Democrats like Sen. Ben Nelson (D-Neb.), Sen. Mark PryorMark Lunsford PryorMedicaid rollback looms for GOP senators in 2020 Cotton pitches anti-Democrat message to SC delegation Ex-Sen. Kay Hagan joins lobby firm MORE (D-Ark.) or even an incoming Democratic senator like West Virginia's Joe ManchinJoseph (Joe) ManchinPavlich: The claim Trump let the mentally ill get guns is a lie Toomey to introduce bill broadening background checks for firearms Scott Walker backs West Virginia attorney general in GOP Senate primary MORE, should he prevail, to cross over to the other side of the aisle.

Those Democrats have sometimes tangled with their party's leaders on big issues like healthcare reform and climate change legislation. Lieberman was almost stripped of a key chairmanship after having supported Sen. John McCainJohn Sidney McCainLawmakers worry about rise of fake video technology Democrats put Dreamers and their party in danger by playing hardball Trump set a good defense budget, but here is how to make it better MORE (R-Ariz.) in the 2008 presidential election.

But there are also major political risks to switching parties. Sen. Arlen Specter (D-Pa.) lost his primary race for reelection after leaving the GOP in 2009. And in a Republican environment in which Tea Party conservatives have unseated centrists in primaries this year, Democratic senators might not be the most eager to join the GOP.

"I think Joe Lieberman is a great patriot," Cornyn said. "He's certainly a hawk on national-security matters."