Homosexuality is illegal in Qatar while a new law has been brought into Russia outlawing the promotion of gay lifestyles.
The task force, which was set up by FIFA president Sepp Blatter earlier this year, meets in Zurich on Thursday and the issue of the homosexuality laws will be on the agenda.
Piara Powar, the executive director of European anti-discrimination group FARE, said it was important the task force did not just deal with racism and will raise the issue of Qatar and Russia's laws.
Powar told Press Association Sport: "The task force is driven by issues of race, quite rightly, but we also have to make sure there is consideration of homophobia and other issues of discrimination.
"Qatar is one of the few countries where homosexuality is still illegal and there are also big challenges in terms of the new law in Russia in regard to the World Cup.
"Qatar wants to host the tournament at the start of a new decade, they will want to present an internationally welcoming face and with FIFA's help we are sure it will be possible to win over the Qataris so that they come into line with the rest of the world, including other countries in the Gulf and Middle East and change the law on homosexuality.
"These are issues of civil rights, fans and players of all races, religions and sexuality need to feel comfortable going to the World Cups in both Russia and Qatar.
"It is going to be quite a challenge but we have to make sure that football becomes the vehicle for social change that we claim it is. This is a big issue."
Powar said that in terms of racism there was much still to do, while the task force should also tackle the issue of the lack of diverse representation in football administration and coaching - and this is a matter that task force chairman Jeffrey Webb also feels strongly about.
Former Football Association chairman David Bernstein will attend the meeting on Thursday, and it is expected to be announced that Tokyo Sexwale, the South African who was imprisoned with Nelson Mandela on Robben Island, and Hilary Shelton, the Washington bureau director of the civil rights group NAACP, will also join the task force.
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