Some Conservatives have also raised concerns about the policy.
Former Tory minister Gavin Barwell said: "It is both morally wrong and unconservative to say that only the wealthiest can fall in love, marry someone and then bring them to the UK."
Tory MP Alicia Kearns told LBC she was "very uncomfortable" with increasing the income requirement for family visas.
"It risks being very unconservative," she said, adding that the Conservatives were "the party of family".
Dr Madeleine Sumption, director of the University of Oxford's Migration Observatory, said the largest impact of the change would fall on those who tend to earn lower wages, including women, younger people and those living outside of London and south-east England.
Labour has said it has "real concerns" about increasing minimum salary requirements for family visas.
On Wednesday a spokesman for Labour leader Sir Keir Starmer said the party wanted to look at the evidence on what impact the change would have, but added: "It looks like it would punish those who fall in love with someone from another country if they don't earn almost £40,000 a year."
Earlier this week, No 10 clarified that the minimum income figure was for a "household as a whole".
The prime minister's spokesman said people could also demonstrate their ability to support a dependant through their level of savings.
He added that the rules included a provision for "exceptional circumstances where there would be unjustifiably harsh consequences" if an individual earning below the threshold could not bring their foreign family member to the UK.
But the spokesman could not give any examples, saying applications would be considered on a "case-by-case basis".
There is uncertainty over whether the new income rules will apply to people who already have a foreign family member in the UK, when their existing visa comes up for renewal.
On Thursday, Downing Street said work was "ongoing" to look at how the change would apply to those renewing visas.
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