A bid for financial guarantees over a home series against South Africa is said to have been backed by ministers, who believe it could help drive support for the Union.
Initially hesitant due to a lack of fans in stadiums, the Cabinet is now believed to support a tour that would see matches played in Scotland, England and Wales and possibly Northern Ireland.
The Scotsman understands Scottish secretary Alister Jack has pushed strongly for the tour behind the scenes.
A UK Government source said: “Alister has very high hopes that it could happen and he’s made his views known at the highest level.
“Seeing the lions play at Murrayfield could be a once-in-a-lifetime opportunity for Scottish rugby fans. He’s desperately keen to see that happen.”
For the series to be financially viable, the Lions would need stadiums to be at 25 per cent capacity.
It is understood if they were full, it could be the most lucrative match at Murrayfield ever.
The Lions have accepted there is no chance of playing in South Africa as planned this summer, so a home series would instead involve four Tests and other games, possibly involving the Barbarians, Japan and the United States.
A Whitehall source said: "Ministers understand that having a home Lions series is not ideal.
"This would not be trying to change the nature of the tour, but recognising these are unprecedented times.
"It's a one-off, so let's embrace it and potentially give a massive boost to pubs and hospitality at the same time."
The UK Government has already indicated support for Euro 2020 matches being able to go ahead in June in England with crowds.
Last week First Minister Nicola Sturgeon offered the same support for matches in Glasgow.
Scotland's road map is due to be published later this month, while the Welsh Government has yet to outline its plans.
The UK Government would not underwrite any matches planned for Dublin and the Irish government has been cautious over the return of crowds.