Mr Mitchell, newly appointed by US President Barack Obama, is to hold talks with Mr Olmert and Palestinian Authority President Mahmoud Abbas, although correspondents say he is not expected to meet Hamas officials.
He has already held talks in Cairo about Egypt's mediation efforts.
Israeli and Palestinian faction representatives have visited Cairo for talks with Egyptian officials since the ceasefires came into effect.
Hamas wants an end of Israel's punishing blockade of Gaza. Israel wants a long-term ceasefire and curbs on Hamas rearming.
During Mr Mitchell's visit, Israelis will want to hear what ideas the US has for advancing the peace process, as well as how Washington will tackle the Iranian nuclear issue, our correspondent says. But with Israeli elections due to take place in two weeks, it is likely the US envoy will spend much of his time listening, as Mr Obama has asked him to do, our correspondent adds.
Mr Mitchell's visit is being seen by many Israelis as a sign of US engagement, and by others as a sign of pressure.
The Gaza Strip's southern frontier is peppered with tunnels into Egypt that were pummelled by air-strikes during Israel's offensive.
One of Israel's stated goals was to halt the smuggling of weapons - including rockets that were being fired against Israeli towns - into the coastal enclave through the network of tunnels.
But smuggling resumed shortly after the non-negotiated cease-fires were declared.
Residents along the border say food, fuel and other goods are moving through the several dozen tunnels that are still operational.
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