Sajid Javid has said he gets abuse "because of my colour" from the "far left" and "far right" of UK politics.
The home secretary told the BBC's Political Thinking podcast he was criticised either for being "too brown" or "not brown enough".
He said he had "sadly got used to" it but had tried to fight it "in every government role I have ever been in".
Mr Javid is among several figures widely touted as contenders to replace Theresa May as Conservative leader.
He has not confirmed his intention to run.
Buzzfeed recently reported that some Conservative Party members had posted messages on Facebook saying they wanted to prevent Mr Javid taking the top job, suggesting the UK was "not ready for a Muslim PM".
The home secretary, whose parents came to the UK from Pakistan, told Political Thinking presenter Nick Robinson he had come to expect social media abuse.
"I get it from the far left, including lots of Asians, who say: 'He's not brown enough.' I get it from the right, and the far right in particular, saying: 'He's too brown,'" Mr Javid said.
"They believe, whether they are coming from the far left or far right, that someone's colour should define who they are - or their background, their faith, or something, that characteristic, rather than the content of their character."
Asked about the posts reported by Buzzfeed, he said: "I think in Britain, anyone who is capable, regardless of whether they are Muslim, or Hindu for that matter, or any religion - or no religion - can be prime minister."
"There are some forces that wouldn't like that but I think the forces against that are much, much stronger. And if you look around the world and you compare Britain to other leading industrial democratic countries, we are way ahead."
He said he had been to European Council meetings of ministers from different member states at which he had been the only non-white person present.
"It's not just me. There are representatives of our country that could say the same thing. That says something about our country, something really, really positive and it's something to celebrate."
He said, while he was happy to talk about faith and religion, "it's not something I define myself by at all and I don't think you should in politics".
He added: "It's not something that should be the determinant of what kind of politician they will or will not be."
Despite winning a vote of confidence in her leadership last December, Mrs May has come under pressure to name an exit date, after the Commons repeatedly rejected her Brexit withdrawal agreement.
In March, she pledged to stand down if and when Parliament ratified her Brexit withdrawal agreement with the EU - but she has not made it clear how long she intends to stay if no deal is reached.
International Development Secretary Rory Stewart and former Work and Pensions Secretary Esther McVey have said they will run to replace her.
Commons leader Andrea Leadsom has said she is "seriously considering" doing the same.
Other widely mentioned possible contenders include Boris Johnson, Michael Gove, Amber Rudd, Dominic Raab, Jeremy Hunt, Penny Mordaunt and Liz Truss.