The most difficult task seems to be finding common ground on background checks, specifically enough common ground that a majority of senators — maybe even a supermajority if some conservatives in the Senate launch a filibuster — agree. Here are the most-talked about proposals and their factions:

Universal background checks

What it would do: What it says it would do is make every gun sale go though a background check. Right now, all sales from a licensed gun dealer (so, say, a sporting goods store) require buyers to go through a background check. But that’s not the case for most Internet sales, sales at gun shows and private sales, such as between friends.

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Supported by: House Democrats. They passed a universal background bill earlier this year; it was one of the first things they did after winning back the House of Representatives. Add to this list the two Democratic leaders of Congress, Speaker Nancy Pelosi (D-Calif.) and Senate Minority Leader Charles E. Schumer (N.Y.). They told Trump this week that it’s this or nothing.

Whom it would alienate: Nearly all Republicans.

The Manchin-Toomey proposal for background checks

What it would do: Expand background checks to include Internet and gun-show sales.

Supported by: The two senators whose names are on the bipartisan bill, Sens. Joe Manchin III (D-W.Va.) and Patrick J. Toomey (R-Pa.). They put together this legislation in 2013 after the Newtown elementary school shooting. It failed in the Senate by six votes, with most Republicans voting against it.

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But now some of those Republicans are open to expanding some background checks, notably Sen. Lindsey O. Graham (R-S.C.), who chairs the key committee for such bills and who has become a Trump confidant, and Sen. Lamar Alexander (R-Tenn.), who chairs another committee more focused on mental health and mass shootings.

Whom it would alienate: Likely both wings of either side, who think it either doesn’t go far enough (Pelosi and Schumer want private sales as part of the deal, too) or goes too far. On the right, there are some who might see any expanded background checks as an unacceptable restriction on guns.

It’s notable that this is the most obvious compromise, and Trump has not yet embraced it despite lobbying from Toomey and Manchin.

Strengthen enforcement of existing background checks

What it would do: There are a couple ways to do this. The Washington Post reported that Attorney General William P. Barr is looking into ways to expedite the death penalty for mass shooters (although many die or expect to die in the first place).

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The Post also reported that Trump was open to an app that would make the background checks that exist more accessible to gun sellers.

Sen. Ted Cruz (R-Tex.), said Sunday that Congress needs to “strengthen background checks” and pushed his 2013 bill to make it a crime to lie on background check forms.

And Graham is working on red-flag legislation to incentivize states to allow a judge to temporarily remove firearms from those whose families deem them a threat to themselves or others.

There is recent precedent for Republicans successfully nibbling around the edges of background checks. In 2018 after mass shootings at a Parkland, Fla., high school, a church in Texas and an open-air concert in Las Vegas, Trump signed into law a bipartisan bill requiring federal agencies and states to more thoroughly report people’s criminal records to the federal background-check system.

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It was a modest effort to make it more expansive, and it didn’t come easy in a Republican-controlled Washington. The legislation passed only when it got folded into a must-pass spending bill.

Supported by: Likely a number of Republicans, who would be relieved if this is what they ended up voting on. It’s politically safer, from the perspective of the National Rifle Association, than expanding background checks. In fact, the NRA supported the 2018 bill.