By Ali Zaslav, Alex Rogers and Ted Barrett, CNN

(CNN) -- Senate Majority Leader Mitch McConnell on Tuesday blocked an effort to quickly pass a measure to increase direct stimulus payments from $600 to $2,000, though the legislation could be voted on at a later time or date if McConnell so chooses.

Senate Minority Leader Chuck Schumer, a Democrat from New York, attempted to pass the change to the omnibus spending bill through a unanimous consent request on the Senate floor, but any senator can halt passage of legislation that way. McConnell, a Kentucky Republican, objected to the request.

The House passed the increase to stimulus payments on Monday with wide bipartisan support. Democratic leaders have cited President Donald Trump's criticism of the $600 payments as too low and the GOP President's push to increase the stimulus checks in their effort to pass the change through the Senate.

The measure could be scheduled for another vote in the Senate at a later date, and what McConnell will do remains unclear as of Tuesday afternoon. Many Republican senators objected to an increase in stimulus payments during negotiations for a Covid-19 relief package throughout the year, citing the growing price tag of the plan. However, multiple GOP senators -- including two members facing runoffs in Georgia next week -- have come out in favor of Trump's push to increase payments, even after the previous legislation passed Congress last week.

McConnell said on Tuesday that the Senate would consider three of Trump's priorities -- further direct financial support for Americans, reexamining Section 230's protections for technology firms and ballot integrity effort -- this week. The reexamining of Section 230 came as Trump vetoed the National Defense Authorization Act over not including revisions to the law that shields internet companies from liability for what is posted on their websites by them or third parties.

McConnell's remark about bringing Trump's priorities "into focus" was not a commitment to bringing votes on the issues. The standoff leaves action on the Senate NDAA override vote in question, and the majority leader might have to file cloture to overcome objections and set up the vote for later this week.

Schumer urged the Senate to join Trump and the House to increase the size of the checks, arguing that "working Americans have taken it on the chin" during the pandemic.

"The fastest way to get money into Americans pockets, is to send some of their tax dollars right back from where they came," he said.

Schumer said $600 is "not enough" for Americans who need the extra money to pay for groceries and rent, and dismissed concerns that the proposal would add too much money to the deficit. He said that Republicans had previously passed nearly $2 trillion in tax cuts and recently fought to include a tax break for corporate meal expenses.

"I don't want to hear that we can't afford it," Schumer said.

If the bill does get a full vote in the Senate, it would need at least 12 Republican senators to join all members of the Democratic caucus in order to cross the 60-vote threshold to advance.

"Those are the three important subjects the President has linked together," said McConnell. "This week, the Senate will begin a process to bring these three priorities into focus."

Vermont Independent Sen. Bernie Sanders who said he would move to delay a vote unless McConnell brings $2,000 stimulus checks to a vote on the floor, continues to push for larger checks.

"The leaders of our country, President Trump, President-elect Biden, Minority Leader Chuck Schumer, the Speaker of the House Nancy Pelosi are all in agreement," Sanders said in floor remarks. "We have got to raise the direct payment to $2,000. So, that is where we are right now in this historic moment. Do we turn our backs on struggling working families or do we respond to their pain?"

The round of stimulus payments at the $600 level included in the $900 billion Covid relief package signed into law Sunday night are expected to start going out this week, an administration official told CNN, while acknowledging that the timing could slide. As with that first round, the new payments will start phasing out for individuals with adjusted gross incomes of more than $75,000, and those making more than $99,000 won't receive anything. The income thresholds would be doubled for couples.