Work and Pensions Minister Damian Hinds said "sought-after" roles in popular industries such as media and fashion were not available to everybody.
On Friday, MPs will debate plans to pay interns at least the minimum wage.
The bill's sponsor, Tory MP Alec Shelbrooke, called unpaid work a "scourge on social mobility".
The proposed National Minimum Wage (Workplace Internships) Bill would require companies to pay interns at least the minimum wage for their work.
Any adult hired as an intern would have to be paid, but the bill excludes school-age children, apprentices and full-time university and college students completing work experience as part of their studies.
Government minister Damian Hinds told ITV's Peston on Sunday programme unpaid internships were not an option for many poorer young people.
Mr Hinds said: "In the media, in fashion, in these very sought-after occupations, these unpaid internships aren't actually accessible to everybody.
"It's important for social mobility that everybody has a crack at getting into the particularly competitive industries."
Mr Hinds's comments echo concerns that unpaid work disadvantages poorer people, who may not be able to rely on family support to cover their living costs.
Mr Shelbrooke said: "Unpaid internships are a scourge on social mobility.
"I'm confident that this government is serious about building a Britain that works for everyone and not just the privileged few so I look forward to government support for my bill."
Responding to the suggestion that having to pay interns might stop employers taking them on, Mr Shelbrooke said: "There are a number of businesses such as KMPG, Ernst & Young and Pimlico Plumbers that already offer paid internships, showing that there really is no excuse for profit-making companies not to pay their workforce."
The government is considering a ban on unpaid internships, amid concerns they give richer candidates an unfair boost in the race to get top jobs.