The 99-year-old, who will turn 100 later this month, set himself the challenge of walking 100 lengths of his garden before his next birthday.
Originally he only wanted to raise £1,000 but as his story spread, the donations poured in and totalled £12.2m by the time he completed the 100 laps on Tuesday 16 April.
A Westminster briefing on Tuesday was told by a Number 10 spokesperson that Captain Moore’s “heroic efforts” would be recognised.
They said: “From his military contributions to his support for NHS staff, Tom has demonstrated a lifetime of bravery and compassion.
“The Prime Minister will certainly be looking at ways to recognise Tom’s heroic efforts.”
Mr Johnson was treated for Covid-19 himself and spent three days in intensive care at St Thomas’ hospital - he is now in recovery at Chequers with his pregnant fiancee Carrie Symonds.
Downing Street said Captain Moore had “captured the heart of the nation” and “embodied the spirit of the whole country in doing their bit for the battle against coronavirus”.
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A petition was launched on Change.org to give Cpt Moore a knighthood in recognition of his charity work during the Covid-19 outbreak.
The petition, which has been signed by 85,000 people in less than four hours, says: "As an NHS midwife, I know how much his fundraising and support means on the ground.
"And how much it means to staff during this awful pandemic. He deserves to be knighted."
Health Secretary Matt Hancock has also called the veteran, who served in India and Burma, an “inspiration to us all”.
On Thursday, the nation will take part in the fourth weekly Clap For Our Carers event as a show of support for care workers across the country.
This week, in light of Captain Moore’s achievement, people have been encouraging others to think of him when they are clapping using the hashtag “#ClapForTom”.
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