BOSTON — Republican Gov. Charlie Baker and Democratic leaders in the Massachusetts Legislature signaled after their first formal meeting Monday that they believe the state’s budget shortfall could be erased without a major showdown over spending or taxes.

Baker, House Speaker Robert DeLeo and Senate President Stanley Rosenberg met for more than an hour in the governor’s office. All three have pledged bipartisan cooperation in the early days of the new administration and seemed at ease with each other during a joint news conference after the meeting.

In his inaugural address Thursday, the new governor said he believed the shortfall in the budget for the fiscal year ending June 30 – which he blamed on a “spending problem” – would exceed $500 million.

The following day he ordered a general hiring freeze in the executive branch of state government.

“We started with a hiring freeze and a contract review because we didn’t want to make the problem any worse than it already was,” Baker said Monday.

Baker said the freeze would not prevent him from making high-level appointments to his new administration, including the yet-to-be-filled cabinet posts of Secretary of Transportation and Secretary of Public Safety.

He did not rule out the possibility of reversing some appointments made in the final weeks of former Democratic Gov. Deval Patrick’s administration.

DeLeo said the meeting represented a good start with the Republican, adding he did not foresee a “battle” between Baker and the Legislature over the deficit.

The governor has ruled out new taxes, and both he and the speaker have ruled out cuts in state aid to cities and towns.

All options should be weighed, Rosenberg said, but he also hoped to avoid local aid cuts and said taxes were always a last resort.

“I don’t think this problem is deep enough that we would have to go to that question,” Rosenberg said. “But from my point of view, everything has to be on the table when you are trying to solve a fiscal problem.”