Sen. John CornynJohn CornynCornyn raises more than M for Senate GOP Is the Biden administration afraid of trade? The Hill's Morning Report - Presented by Facebook - After high drama, Senate lifts debt limit 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.

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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 PryorBottom line Everybody wants Joe Manchin Cotton glides to reelection in Arkansas MORE (D-Ark.) or even an incoming Democratic senator like West Virginia's Joe ManchinJoe ManchinK Street revenues boom Biden champions economic plan as Democrats scale back ambitions On The Money — Democrats eye tough choices as deadline looms 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 McCainMeghan McCain: 'SNL' parodies made me feel like 'laughing stock of the country' Our military shouldn't be held hostage to 'water politics' Meghan McCain blames 'toxic' hostility for 'The View' exit 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."