WASHINGTON (Reuters) - The U.S. Senate this month will debate legislation that passed overwhelmingly in the House of Representatives and aims to give Puerto Rico some relief from its $70 billion debt, Senate Majority Leader Mitch McConnell said on Tuesday.

McConnell's announcement came after some House members briefed fellow Republicans in the Senate on the content of the legislation, which would establish a federal oversight board to restructure Puerto Rican debt.

Backers of the House-passed bill are hoping the Senate signs off on the legislation before July 1, when Puerto Rico faces a deadline on a $1.9 billion debt payment.

Representative Rob Bishop, who chairs the House Natural Resources Committee, which steered the legislation through that chamber, told reporters that Republican senators peppered him with questions about the makeup of the oversight board and its powers. He added that he assured senators the bill is not a taxpayer bailout of Puerto Rico and that it would not have an impact on U.S. states.

Puerto Rico is a U.S. territory and some critics of the bill have expressed concerns that a control board appointed by Washington might not have the Caribbean island's best interests in mind.

The legislation also got a boost, albeit a tepid one, on Tuesday when Senate Finance Committee Chairman Orrin Hatch offered qualified support for the House bill. He added that he was considering offering changes to the bill but did not elaborate.

Hatch's Finance Committee has oversight of the Puerto Rico debt legislation.

Some Senate Democrats also have expressed reservations with the bill and indicated that they could seek amendments, but it was not clear if they actually would.

Any changes made by the Senate to the House-passed bill would send it back to the House to accept, reject or make further amendments.

Bishop left open the possibility of July 1 arriving with Congress still working on the Puerto Rico bill.

"It's not the end of the world," Bishop said of the possibility of Congress putting finishing touches on legislation into July.

"Market forces, as long as they realize we're moving forward to a solution and we're close to it, I think that would mitigate against anything that would be negative that would impact the entire market," Bishop added.

The Obama administration has voiced its support for the House bill, which passed on a strong, bipartisan vote of 297-127.

Puerto Rico, which suffers from a poverty rate of around 45 percent, has been hobbled by its worsening debt problems, with some schools and medical facilities closing and island residents fleeing to the mainland United States where many find jobs, primarily in central Florida.

(Reporting By Richard Cowan; Editing by David Gregorio and Dan Grebler)