The club's plans to give the Argentina striker a medical on Wednesday morning were blocked by West Ham and United are taking the issue to FIFA's dispute resolution chamber.
Gill said the case would be rushed through by FIFA so that it could be dealt with as soon as possible.
He also expressed his confidence that Tevez would eventually become a United player.
With no end in sight to the row over Tevez's registration, the 24-year-old, who has scrapped plans to have a medical in Manchester this week, has called in the world governing body in a bid to resolve the issue once and for all.
A special 'Dispute Resolution Chamber' will be set up, possibly within a fortnight, to effectively determine who currently owns Tevez, West Ham or his adviser Kia Joorabchian.
"There has been a lot of discussions over the last few days between Manchester United, West Ham and the Premier League but the case will probably go before FIFA's dispute resolution chamber," said Gill.
"We are very confident this will be resolved in favour of the player and he will be free to achieve his wish of joining Manchester United."
West Ham confirmed they have written to the Premier League on Wednesday morning requesting the matter be referred to FIFA.
A Hammers spokesman said: "It now makes sense for this to be dealt with by FIFA.
"All parties agree this is the best way forward and of course we will be involved in helping this be resolved as quickly as possible."
Talks between United lawyer Maurice Watkins and West Ham chairman Eggert Magnusson at the Premier League's offices in London on Tuesday ended in stalemate with no agreement reached over the 23-year-old.
Tevez flew into Manchester on Tuesday evening with the intention of undergoing a medical on Wednesday morning but West Ham refused to grant permission.
A West Ham spokesman said: "Meetings have taken place between the two clubs but no agreement has been reached in relation to Carlos Tevez."
One club cannot give a player a medical unless his current club agrees and West Ham will not give formal permission for a medical until they have been made an offer and accepted it.
West Ham remain adamant that they hold Tevez's registration, that he is contracted to them for another three years and that they have unilaterally terminated all third-party agreements with Tevez's representative Kia Joorabchian.
Joorabchian maintains that his company Media Sports Investments (MSI) own the economic rights to the player, and that they have already agreed personal terms with Manchester United.
United are understood to have agreed an initial loan deal for Tevez with a view to a permanent transfer, a similar deal to that which took Javier Mascherano from West Ham to Liverpool in January.
That hit a stumbling block however because in January, West Ham cancelled Mascherano's registration. The Premier League are not allowing the club to do the same procedure with Tevez on the basis that such a move would indicate the original agreements with Joorabchian were still binding despite assurances to the contrary.
United expect a FIFA panel will hear the Tevez case in around two weeks. The world govening body, however said they will have to first decide whether they are the correct organisation to deal with it.
FIFA spokesman Andreas Herren said: "If a complaint is submitted to FIFA our legal department will take a look at it and see whether we are the competent authority to deal with the matter."