More than 100,000 people have offered homes to Ukrainian refugees in the first 24 hours of a government scheme that allows families and individuals to bring them to the UK.

The website for registering interest in the scheme crashed for a short while because of the numbers offering help.

On Tuesday afternoon, the Department for Levelling Up, Housing and Communities tweeted: “Thanks to the generosity of the British public we’ve received over 100K expressions of interest from individuals and organisations so far in our Homes for Ukraine scheme.”

The Foreign Office minister, James Cleverly, said earlier “10,000 people every hour” were signing up.

A spokesperson said the Homes for Ukraine website “temporarily stalled” after it went live late on Monday afternoon, owing to the “enormous generosity of the British public”. He said the total number of offers was continuing to rise, with offers doubling within hours.

Almost 3 million people have fled Ukraine since the Russian invasion, with more displaced within the country as cities are shelled.

Until Monday, only Ukrainians with family links to the UK could apply for visas that would give them access to work and benefits.

But on Monday, the levelling up secretary, Michael Gove, announced that sponsors could bring any Ukrainian to the UK to live in their home for a minimum of six months, for which they would receive a £350 monthly payment.

Under the scheme, refugees would be allowed to live and work in the UK for up to three years and receive full and unrestricted access to benefits, healthcare, employment and other support.

Cleverly said he was proud that the numbers of people wanting to help Ukrainians meant the website had crashed.

“Frankly, I’m glad we moved quickly on this and we’re moving quickly to ensure we’re able to help the Ukrainian refugees,” he said. “I know this is a weird thing to say as a government minister – I’m glad the website crashed, because it is a reflection of that generosity of the British people.”

Asked whether he would be taking part in the scheme, Cleverly said: “I have genuinely considered this. I’ve discussed this with my wife. I don’t know whether our personal circumstances will allow us to do this right at the moment.”

The minister said his wife was going through medical treatment but added:it was “absolutely something that I’m considering”.

The administration of the scheme has been criticised by Labour as families need to identify a Ukrainian family they wish to sponsor.

Lisa Nandy, the shadow levelling up secretary, called it a “DIY scheme” and said of Gove, who set out details of the scheme in the Commons on Monday: “He can’t seriously be asking Ukrainian families who are fleeing Vladimir Putin, who have left their homes with nothing, to get on to Instagram and advertise themselves in the hope that a British family might notice them.”

Cleverly said local authorities would get the help they needed. “We don’t know how many will seek refuge and for how long,” he told BBC Radio 4’s Today programme.

He said charities and faith groups would lead on matching people with refugees seeking to come to the UK, adding that the government taking over the matching would be a “slow and bureaucratic process”.