In a letter, Gove and Johnson have challenged Cameron to guarantee that Turkey will not join the EU by pledging the UK's veto.
The letter comes as the pro-Brexit campaign has focussed more and more on Turkey's hypothetical EU membership.
Campaigners backing Leave have said that it is government policy to back Turkey's membership.
"Despite the rapidly accelerating pace of accession negotiations, In campaigners maintain that Turkey 'is not an issue in this referendum and it shouldn’t be'," the letter said.
"Others assert that the UK has 'a veto' on Turkish accession. This claim is obviously artificial given the Government’s commitment to Turkish accession at the earliest possible opportunity."
All EU nations have a veto on the accession of other nations, meaning in theory the UK could block Turkey from joining the 28-member bloc.
Gove and Johnson also want the government to block visa-free travel for Turkish citizens. They added: "If the government cannot give this guarantee, the public will draw the reasonable conclusion that the only way to avoid having common borders with Turkey is to Vote Leave and take back control on 23 June."
However, pro-Remain ministers Theresa May and Philip Hammond said earlier this week: "The government's policy is, and will remain, to maintain current visa requirements for all Turkish nationals wishing to visit the UK, regardless of what arrangements other Member States in the Schengen area may make with Turkey.
"Schengen visas do not give anyone the right to access the UK."