Mayor Bill de Blasio, seen with Gov. Andrew Cuomo at an unrelated new conference, pressed for quick approval of the increased minimum wage. Eduardo Munoz/Reuters

New York City Mayor Bill de Blasio on Tuesday pushed the state Legislature to approve an increase in the state minimum wage “whenever it first can be done,” a request that comes as labor advocates and some elected officials encourage Gov. Andrew Cuomo to call a special legislative session before the year’s end.

Beginning Jan. 1, Republicans will hold a majority in the state Senate, placing in jeopardy several Democratic policy initiatives, including a higher minimum wage.

Some Democrats are now calling on Mr. Cuomo to summon the Legislature back to Albany next month before the GOP Senate takeover, but the newly re-elected Democratic governor has resisted.

“I certainly look forward to talking to the governor about whether there will or will not be” a special session, Mr. de Blasio said.

The mayor, who endorsed Mr. Cuomo for re-election, declined to say directly whether the governor should call a special session, instead simply noting, “We need to pass the minimum wage whenever it first can be done.”

Mr. Cuomo’s reluctance to call a special session has led some of his fellow Democrats to question his commitment to many of the left-leaning policy initiatives he endorsed during the campaign.

In a radio interview last week, Mr. Cuomo said he doesn’t believe a special session is needed.

“Given that there is going to be such a dramatic turnover, basically on the Senate side, they’ll need some time to orient their new members,” he said.

On Tuesday, an aide to Mr. Cuomo said: “We have not heard from any legislative leader that they have the votes or desire to pass anything.”

A spokesman for GOP Senate leader Dean Skelos didn’t respond to a request for comment nor did a spokeswoman for Sen. Jeff Klein, the Democrat who co-leads the Senate with Mr. Skelos.

A spokesman for Democratic Assembly Speaker Sheldon Silver said there has been no discussion, to date, about a special session.

The Senate is led by a bipartisan coalition that negotiates which bills make it to the floor. In the past, Mr. Skelos has opposed an increase in the minimum wage and Mr. Klein has supported it.

In the spring, as part of a deal to gain the backing of the Working Families Party, Mr. Cuomo endorsed increasing the minimum wage to $10.10 an hour and allowing some jurisdictions, such as New York City and Nassau and Westchester counties, to raise the local minimum wage as much as 30% higher than the statewide level.

Mr. Cuomo had opposed letting local governments set their minimum wage.

The state minimum wage of $8 an hour is slated to increase to $8.75 on Dec. 31 and $9 on Dec. 31, 2015.

Michael Kink, director of Strong Economy for All Coalition, a group of unions and community organizations that supports a higher minimum wage, said the governor should call a special session. He suggested packaging a higher minimum raise with pay raises for lawmakers to attract bipartisan support.

“The three million New Yorkers waiting for a raise could use one as soon as possible,” he said. “A special session right now is the fastest and arguably the best way to get it done.”

State Sen. Liz Krueger, a Manhattan Democrat, agreed.

“I would love the opportunity to come back to Albany and vote for minimum wage and a number of other bills that I believe the governor is committed to, the Assembly is committed to, and the Senate Democrats are committed to,” she said. “That will be much tougher next session, given the election results.”

Many business groups and Republicans oppose an increase, arguing the higher labor costs would lead employers to reduce hiring.

“Minimum wage hikes reduce jobs and hurt the most vulnerable of the population, who are looking for their first jobs, and that’s why we are opposed to them,” said Ed Cox, chairman of the state GOP.

The Albany legislative session opens in January and ends in June, but the Legislature has met at least once between August and December every year from 2001 to 2011, according to records from the government watchdog New York Public Interest Research Group.

In 2011, Mr. Cuomo’s first year in office, the governor called a special session to increase taxes to help close a budget gap.

Next year in the Senate, there will be 32 Republicans and 31 Democrats. If a special session is held next month before the GOP takeover, there could be as many as 33 Democratic votes, but it is unclear whether Senate Democrats—a divided group—could pull together.

Sen. Ruben Diaz, a Bronx Democrat who sometimes votes with Republicans, said Tuesday he was undecided on a minimum-wage increase and didn’t see the urgency for an off-season session.

This year, Mr. de Blasio led an unsuccessful charge to help Democrats win control of the state Senate. The mayor now faces an uphill battle to advance his Albany policy agenda, which includes public financing of state political campaigns and a measure that would make state financial aid available to illegal immigrants.

“There’s a very serious agenda that has to be achieved in Albany, and I think everyone should be judged on how they respond to that agenda,” he said Tuesday.

Mr. Cuomo endorsed these policy initiatives as part of his deal with the Working Families Party, but some political observers say the governor didn’t work as aggressively as the mayor to support Democratic Senate candidates.

Next year, some say, the governor could use the Senate GOP’s leadership position’s opposition to the liberal policies as a cover if they fail.

Write to Mike Vilensky at mike.vilensky@wsj.com and Michael Howard Saul at michael.saul@wsj.com