It followed crunch meetings between Prime Minister Theresa May and Brexit Secretary David Davis, who insisted a cut-off date be included.
The proposal would see the UK match EU trade tariffs temporarily in order to avoid a hard Irish border post-Brexit.
Brexiteers want to ensure the backstop could not continue indefinitely.
Responding on Twitter, EU chief negotiator Michel Barnier set out the criteria on which he would judge the UK's proposal, including the need for a "workable solution" to avoid a hard border in Northern Ireland.
The UK is due to leave the EU in March 2019, and the government is trying to make progress before a crucial meeting of EU leaders later this month.
BBC political editor Laura Kuenssberg said Mr Davis had "pushed back very hard" against the proposals on Wednesday and had two meetings with the prime minister on Thursday before a deal was agreed.
A source close to Mr Davis told the BBC after the second meeting that "the paper has been amended and now expresses in much more detail the time-limited nature of our proposal".
According to the document - which has yet to be agreed with the EU - the "temporary customs arrangement", if it is needed, would be "time-limited".
A long-term "future customs arrangement" will be in place "by the end of December 2021 at the latest", it says.
Mrs May also held meetings earlier with Foreign Secretary Boris Johnson and International Trade Secretary Liam Fox.
Her official spokesman described the meetings as "constructive" and - with speculation Mr Davis might quit over the issue - was asked if anyone had threatened to resign. He replied: "No."
The UK has said it will leave the EU's customs union, which allows trade within the EU without any tariffs or many border checks.
The UK and the EU are yet to agree how trade in goods will operate after Brexit - but they have said that a "backstop" option is needed in case no deal is done, or the technology is not ready in time, to avoid the return of a hard border between Northern Ireland and the Republic of Ireland.
The UK has said that the EU's initial "backstop" proposal - effectively keeping Northern Ireland in the customs union - would create what amounted to a border between Northern Ireland and the rest of the UK and was not acceptable.
Instead, the UK is proposing a backup plan which would see the whole of the UK temporarily aligned with the EU's customs union after December 2020 - when the 21-month post-Brexit transition period ends.
The plan, which Theresa May has said would only apply in a "limited set of circumstances", would see the UK match EU tariffs in order to avoid border checks.