Downing Street has refused to condemn controversial past remarks on pregnancies, eugenics and race reportedly made by a new adviser.
Labour said Andrew Sabisky should be sacked for suggesting black people had lower average IQs than white people and compulsory contraception could prevent "creating a permanent underclass".
No 10 did not officially confirm his appointment or any role he may have.
The SNP and Liberal Democrats also said Mr Sabisky should go.
Mr Sabisky, appointed after the PM's chief adviser Dominic Cummings called for "misfits and weirdos" to apply for jobs in Downing Street, has been contacted by the BBC for comment.
When asked on Monday, Downing Street did not comment on the remarks attributed to Mr Sabisky.
Boris Johnson's spokesman added: "The prime minister's views on a range of subjects are well publicised and documented."
Lobby briefings see journalists pile into a room to ask the Prime Minister's official spokesman whatever they like.
It doesn't mean that the spokesman - or in today's case the deputy - will give a fulsome answer. And certainly, today, one was not forthcoming.
The spokesman was asked about the PM's views on eugenics. On race. On women's sport. Repeatedly.
Rather than explicitly condemn Andrew Sabisky's reported remarks the response was to point to Boris Johnson "well documented" views.
But it proved to be an inadequate effort to deflect the story; adding fuel to the fire rather than dampening it down. And, of course, Boris Johnson has been known to attract criticism for some of his own controversial comments - on a variety of subjects - in the past.
The normal laws of politics might dictate that Andrew Sabisky is in line to lose the job he's so recently attained.
But this Downing Street operation has shown that it isn't afraid of a bit of controversy. It doesn't necessarily feel bound by the 'rules'. And Dominic Cummings has, at times, appeared to enjoy outraging the establishment.
That isn't to say the views apparently expressed by Andrew Sabisky are in anyway shared by his new colleagues at Number 10. And it may be that today's hesitant and tight-lipped response was a clumsy effort to buy time rather than a decisive show of defiance.
However if they don't sack Andrew Sabisky, that is exactly what it will be.
In response, Conservative MP Caroline Nokes tweeted: "Cannot believe No 10 has refused to comment on Andrew Sabisky. I don't know him from a bar of soap, but don't think we'd get on... must be no place in government for the views he's expressed."
In a comment on a 2014 blog post on Mr Cummings' website, made by a user called "Andrew Sabisky" that used the same picture as his Twitter page, it is suggested that compulsory contraception could be used to stop a "permanent underclass".
"One way to get around the problems of unplanned pregnancies creating a permanent underclass would be to legally enforce universal uptake of long-term contraception at the onset of puberty," says the post.
"Vaccination laws give it a precedent, I would argue."
In a comment on another blog post on a different website in 2014, what appears to be the same user suggested black Americans had a lower average IQ than white Americans.
In a comment on a different blog that same year, a user with his name said: "There are excellent reasons to think the very real racial differences in intelligence are significantly - even mostly - genetic in origin, though the degree is of course a very serious subject of scholarly debate."
Mr Sabisky also suggested to Schools Week in July 2016 that the benefits of a purported cognitive enhancer, which can prove fatal, are "probably worth a dead kid once a year".
"Eugenics are about selecting 'for' good things," he said in the same interview. "Intelligence is largely inherited and it correlates with better outcomes: physical health, income, lower mental illness.
And in a Twitter post from 2019, he said: "I am always straight up in saying that women's sport is more comparable to the Paralympics than it is to men's."
Labour Party chairman Ian Lavery said: "It is disgusting that not only has Number 10 failed to condemn [these] appalling comments, but also seems to have endorsed the idea that white people are more intelligent than black people.
"Boris Johnson should have the backbone to make a statement in his own words on why he has made this appointment, whether he stands by it, and his own views on the subject of eugenics."
Labour's shadow Cabinet Office minister Jon Trickett said: "[Mr Sabisky] must of course be removed from this position immediately."
Acting Liberal Democrat leader Ed Davey urged the prime minister to "put an end to the offence caused and sack Andrew Sabisky".
"This Conservative government is a national embarrassment," he said. "By giving Dominic Cummings such power and then failing to control him, Boris Johnson is revealing who really is in charge."
And Scotland's First Minister Nicola Sturgeon said the government must "demonstrate some basic but fundamental values", tweeting: "These are really not acceptable headlines for any government to be generating."
When asked about the remarks on Sky News, Environment Secretary George Eustice said it was a "matter for Dominic Cummings and Number 10".
On Sunday, Transport Secretary Grant Shapps told the BBC: "I don't know the individual but they are particularly not views that I or the government shares in any way, shape or form."