WASHINGTON — A group of Senate Republicans has unveiled a plan they said would spur hiring and revive the flagging economy, the latest sign that President Barack Obama’s full-throated attacks over jobs may be leaving wounds.

The new bill was outlined by a cross section of lawmakers and was notable not for its fresh policy approach, but for its clear admission that the party feared losing the rhetorical fight over job creation.

Republican leaders in the Senate have for months put the onus on the White House to outline and sell a strategy for economic recovery — and then blasted the president and his fellow Democrats for not pushing a bill that could win bipartisan support.

GOP lawmakers have offered a raft of proposals but have resisted presenting them as a single bill that could be evaluated, attacked and — likely — killed in the politically divided Congress.

But restlessness with that approach is clearly growing. Obama has been touring the country promoting his jobs bill and repeatedly calling on Congress to “pass the bill.”

“We just thought it was time to put this all into a package. I will freely admit to you that part of it is in response to the president saying we don’t have a proposal,” said Sen. John McCain, R-Ariz. McCain led the effort to write the legislation.