A trans Atlantic rift over swaps could be ended by an offer from the head of the U.S. derivatives regulator that would spare foreign banks from some tough U.S. rules and from incurring excessive costs, according to two people briefed on the matter.

Gary Gensler, head of the U.S. Commodity Futures Trading Commission, triggered a rift with Europe last year by insisting that foreign companies should comply with the agency's rules if they wanted to trade risky derivatives with U.S. firms.

European regulators, banks and even some leaders within the CFTC complained about this aggressive stance, saying the agency should instead recognize similar foreign rules, as long as they are comparable.

Big foreign banks such as Deutsche Bank AG , Credit Suisse Group AG , Barclays Plc and others feared they would lose a global competitive edge if they had to deal with the cost of both U.S. rules and those from their home country.

Gensler has since drafted a compromise "cross-border" approach, one of the people said, that includes phased-in compliance and would potentially spare foreign banks from some of the toughest U.S. rules.

It is this compromise that Gensler is trying to sell to Europe, ahead of a Friday CFTC meeting on the matter.

"For now, it's looking good," a second person said on the talks between Gensler and two top aides of Michel Barnier, European Union financial services czar. "It could be done by (Monday) night or on Tuesday."

The two sides are working to boil down the compromise to fit on one page, to ensure its clarity and simplicity, this person also said. The CFTC declined to comment. Barnier's office had no immediate comment.

The stakes are high, with banks heavily lobbying against new rules to rein in the $630 trillion market, dominated in the United States by banks such as Citigroup Inc , Bank of America Corp and JPMorgan Chase & Co .

The rules, which include trading requirements such as registration and data reporting, would make swaps markets less opaque after the credit meltdown and prevent risk affecting U.S. companies from building up abroad.

The compromise being discussed among Gensler and Barnier's aides involves the platforms on which trading takes place, allowing foreign firms some options when trading with U.S. clients and removing some of the toughest regulatory requirements, the two people said.

Both stressed that talks are continuing and a deal could still fall through. Also, Gensler still needs to secure two other votes from fellow CFTC commissioners. The hope is that a compromise deal with Europe could pressure the other officials to sign off on the cross-border guidance.

In one example of how such a compromise would work, a non-U.S. bank could opt to comply with U.S. or European law depending on whether it executed a trade on a platform in America or abroad, and provided that European rules for such platforms were in place.

In another example, the trade could be executed on a European platform that was already registered with the CFTC as a so-called Foreign Board of Trade (FBOT).

"That's one way that we've agreed to, the only thing is we're not sure that goes far enough," the first person said, adding other problems were also still outstanding.

In both cases, the bank's own requirements such as data reporting, registration with authorities and putting in place clearing requirements, would be less cumbersome.

(Reporting by Douwe Miedema; Editing by Karey Van Hall and Andre Grenon)