The news comes as Israeli and Hamas officials discuss a ceasefire through Egyptian mediators after days of violence in Gaza that have left 125 Palestinians dead in Israeli operations.
The vice-president's visit follows a brief trip to the region by Condoleezza Rice, the US secretary of state, who held meetings with Mahmoud Abbas, the Palestinian president, and Ehud Olmert, the Israeli prime minister, in order to press for the resumption of peace talks.
Bush said that Cheney would also be seeking assurances that "the relevant parties ... uphold their obligations under the road map".
Cheney is also expected to push for an increase in oil output with Saudi Arabian officials as prices continue to reach record levels, Dana Perino, the White House spokesperson, said on Monday.
"I'm sure that energy issues will come up ... obviously we want to see an increase in production," she said.
Bush visited Saudi Arabia in January and urged the Organisation of Petroleum Exporting Countries (Opec) - of which Saudi Arabia is a key member - to increase production, although the group decided not to boost output.
Cheney's visit to Turkey, meanwhile, will likely focus on Ankara's tensions with Baghdad following its recent military offensive against Kurdish rebels in the north of Iraq.