Britain could rewrite its Internal Market Bill, now passing through parliament, as part of a Brexit deal with the European Union, Bloomberg news said in a tweet on Sunday.
He added: "I want a deal (with the EU) but it takes both sides to compromise in order for there to be one".
More than 70 British business groups representing over 7 million workers made a last-ditch attempt on Sunday to persuade politicians to get back to the negotiating table next week and strike a Brexit deal.
Officials in Brussels say Mr Barnier is still planning to travel tomorrow to London for further discussions, despite warnings from Downing Street that he might not get the warmest of welcomes.
It appeared that Mr Johnson had shelved plans for further negotiations altogether, after the UK's Brexit chief negotiator Lord Frost last week told his European Union counterpart Michel Barnier not to come to London for planned talks without a major change in position from European Union leaders.
Mr Johnson previously set a deadline for an agreement last week.
Lord Frost and Mr Barnier are due to hold a call early this week and Mr Gove will meet his counterpart on the UK-EU joint committee, Maros Sefcovic, in London on Monday.
Asked if he was walking away from talks, Johnson said: "If there's a fundamental change of approach, of course we are always willing to listen, but it didn't seem particularly encouraging from the summit in Brussels".
"We made clear we needed to see a change in approach from the European Union", he said, adding that Barnier would be speaking to his United Kingdom counterpart David Frost in the coming days.
The Prime Minister and Mr Gove will call businesses with large supply chains and representative organisations on Tuesday, as the Government launches an advertising drive to warn "time is running out".
At what was supposed to be the "Brexit summit" on Thursday, the European Union delivered an ultimatum: it said it was concerned by a lack of progress and called on London to yield on key sticking points or see a rupture of ties with the bloc from January 1 after the end of the Brexit transition period.
The EU has said it is prepared to "intensify" the talks but it won't agree a deal at "any price".
European Commission President Ursula von der Leyen said she was keen for a deal, though Macron was more downbeat.
Talks between the European Union and Britain have been stalled for weeks due to disagreements on key issues, such as the EU's access to British fishing areas, equality in the marketplace and governance.
Speaking at the end of the summit in Brussels on Thursday, European Council president Charles Michel said the EU was ready to carry on with negotiations.