The Obama administration on Friday sent a message to Republican governors who have turned down federal funding for high-speed rail systems, putting $2.4 billion that Florida rejected up for grabs to states that want to invest in the trains.

Florida Gov. Rick Scott, a Republican who won election in November, turned back funding from a high-speed rail line that would have linked Tampa to Orlando, saying he feared the state would be saddled with cost overruns.

Two other newly elected GOP governors — John Kasich of Ohio and Scott Walker of Wisconsin — also rejected high-speed rail funding that had been lobbied for by the governors they replaced.

Transportation Secretary Ray LaHood said he would make the $2.4 billion available to other states through a competitive process that would reward those states most likely to move swiftly to put people to work building the rail service.

“States across the country have been banging down our door for the opportunity to receive additional high-speed rail dollars and to deliver all of its economic benefits to their citizens,” LaHood said in a statement.

LaHood described high-speed rail as the administration’s signature transportation project last month in announcing a six-year proposal to spend $53 billion to construct a national high-speed network and rail lines that would feed into it.

The timing of the investment has been challenged by Republicans, who contend that expensive initiatives should be put on hold until the deficit is reduced. In the debate this week over current fiscal-year spending, Senate Democrats proposed a $1.5 billion reduction in funding for high-speed rail, and House Republicans sought to cut the amount by $2.5 billion.

On Thursday, Florida Rep. John L. Mica (R) said that he sought to broker a deal that would have allowed private investors into the Florida rail proposal as a hedge against overruns but that Scott could not be persuaded to support what he called a “boondoggle.”

The rejected plan called for trains that would have traveled as fast as 170 miles per hour on the 85-mile route from Tampa to Orlando.

Maryland Gov. Martin O’Malley and several members of Congress have written to LaHood seeking the funds for rail upgrades along the congested Northeast Corridor.