The US delegation - which also includes former secretary of state Colin Powell and vice-chairman of the Joint Chiefs of Staff, General Peter Pace - arrived on Friday, two days after most foreign leaders came to pay their respects.
A Saudi official said Cheney would have lunch with Abdullah, who formally succeeded Fahd after an allegiance ceremony at which thousands of Saudis pledged loyalty and support to their new monarch.
Bush won Fahd's agreement to send half a million troops to Saudi Arabia in 1990 to launch the war on Iraq that forced it to end its occupation of Kuwait.
The presence of non-Muslim soldiers in Saudi Arabia, home to Islam's holiest mosques at Makka and Madina, has angered loyalists of Saudi-born al-Qaida leader Osama bin Laden.
Most of the foreign troops were withdrawn in 2003.
Relations between the United States and Saudi Arabia, a key US ally and major oil supplier, were shaken by the 11 September 2001 attacks on US cities, carried out mainly by Saudi aircraft hijackers.
Saudi Information Minister Iyad al-Madani said on Tuesday the two countries had had a "bumpy ride" since 2001, but were reaching a "fuller understanding of how things function on either side of the ocean".