White House budget director Mick Mulvaney on Wednesday said the White House hopes to send Congress a package of new spending cut proposals in the coming weeks, even as Senate Majority Leader Mitch McConnell appears cool on trying to roll back parts of the recent $1.3 trillion spending bill.

Some House Republicans are bullish on putting together a so-called “rescissions” package to try to claw back some of the spending in the bill, and Mr. Mulvaney said the White House hopes to work with lawmakers in the coming months on a package.

“[We] hope to have something here in the next couple weeks,” Mr. Mulvaney told a House appropriations subcommittee Wednesday.

The package of proposed cuts could be as much as $60 billion, according to reports.

Even as President Trump signed the measure last month, the president decried some of the non-military spending demanded by Democrats as wasteful and said he would never sign a similar bill again.

Mr. Mulvaney said the package wouldn’t necessarily be tied to items in the spending bill. The proposed cuts would require simple majority support in the House and Senate to take effect beyond a 45-day window.

But Mr. McConnell, who helped negotiate the final details of the $1.3 trillion package, was cool on the idea of trying to roll back parts of it.

Mr. McConnell on Tuesday pointed out that the president and his staff were involved in the negotiations, and that Mr. Trump did sign the bill into law.

“You can’t make an agreement one month … and come back the next month and say, oh, we really didn’t mean our agreement,” Mr. McConnell said on Fox News.

But House conservatives upset with the higher domestic spending levels in the budget deal say they want to press forward.

“This is a useful tool,” said Rep. Tom Graves, Georgia Republican.

But Rep. Nita Lowey, the ranking member on the House Appropriations Committee, said she was “baffled” by the push, saying the White House already had plenty of opportunities to weigh in during the negotiations.

“This whole exercise seems to me a bit bizarre,” said Ms. Lowey, New York Democrat.

Mr. Mulvaney said the White House had known about the general topline budget numbers from a previously agreed-upon two-year deal, but that there wasn’t much time to review the actual spending bill text before lawmakers voted on the measure.

“You didn’t hear the president complain about the topline numbers — you heard him complain about some of the individual items, and that’s what rescissions are designed to go after,” he said. “We don’t think this is unusual; we don’t think it’s something that’s unprecedented; we certainly don’t think it’s something that indicates going back on any type of agreement.”