European leaders gathered in Brussels seemingly have a new idea to try to break the current stalemate in Brexit negotiations, but it could prove costly for Prime Minister Theresa May and the U.K.

May is reportedly willing to extend the transition process. This is the portion of time after the U.K. has officially left the EU but when it is allowed to negotiate its future relationship, including a new trade deal with the rest of the EU. It will also give companies and people on both sides time to prepare for the new arrangements. This is currently set to last 21 months, between the exit day on March 29 next year and the end of 2020.

The idea behind the proposal for the extension is to have enough time to negotiate a trade deal with the rest of the EU that will avoid having to use the so-called Irish backstop. This backstop is a future proposal that both the EU and the U.K. need to agree on to avoid a hard border between Northern Ireland and the Republic of Ireland. It will only be triggered if both sides do no reach a future trade deal. The EU and the U.K. want to have a backup solution in case that ends up being the case.

This is the issue that is currently blocking progress in Brexit talks.

Danielle Haralambous, U.K. analyst at the Economist Intelligence Unit, told CNBC via email: "With the current phase of negotiations in deadlock over the Irish border issue, an extension of the transition period could in theory provide more time to negotiate a trade arrangement that permanently solves it. But this will be a hard-sell to the Brexiteers (those U.K. politicians that have campaigned to leave the EU)."

"They will reject the prospect of another year of financial contributions to the EU and adherence to EU rules," she added.