WASHINGTON — (CNN) Senate Majority Leader Mitch McConnell is quietly trying to make it harder for Democrats to block bills to fund the government.

In private conversations with Democrats in the thick of budget negotiations this week, McConnell proposed a plan that would prevent senators from blocking debate on spending bills approved with bipartisan support, according to two people familiar with the talks. While senators could still filibuster an effort to end debate under the McConnell plan, the idea was to speed up the deliberative body and put pressure on Senate Democrats to ultimately relent on their disagreements over spending levels.

Senate Minority Leader Harry Reid strongly objected to the proposal, sources said.

McConnell, however, is still trying to move ahead. He has tasked Senate Rules Chairman Roy Blunt, R-Missouri, and others with reviewing a possible change in how many votes are needed to block appropriations bills.

The issue is significant because McConnell has come under unrelenting pressure from House conservatives to change filibuster rules — something bound to intensify in December when Congress will be forced once again to fund the government or risk a possible shutdown. McConnell, so far, has refused to employ the “nuclear option” — that would change filibuster rules by a simple majority of senators — meaning that any rules change would need a supermajority of 67 senators to enact, a high threshold.

Don Stewart, a McConnell spokesman, declined to comment on private conversations between members.

On Wednesday, McConnell’s chief deputy, Sen. John Cornyn of Texas, said it was imperative for Republican senators to begin the process of changing the rules on funding bills in order to expedite debate in the slow-moving body.

McConnell asked the group “to look at changes in the 60-vote cloture requirement, particularly as in regards to appropriations,” Cornyn said Wednesday.

“I think there’s a big concern that I think this deal goes a long way to help alleviate, by getting those caps into next year. But if the minority can basically force us into an omnibus or (a continuing resolution) by just filibustering the individual appropriation bills, that’s really pretty damaging to the institution and our ability to conduct business,” Cornyn said. “So there is a serious look being taken to that.”

Reid said Tuesday during a meeting with reporters that he and McConnell had discussed rules changes.

“There were a number of difficult issues dealing with procedures here in the Senate,” Reid said of their talk.

But asked specifically if they talked about changing the number of votes needed to filibuster spending bills, Reid only said, “We had a lot of discussions on Senate procedure.”

The move, if Senate Republicans go forward with it, would effectively make it harder for Senate Democrats to filibuster each single appropriation bill, which they had done to date because of disagreements over spending levels.

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