But Olmert has insisted Israel would open Gaza's borders if Hamas released Gilad Shalit, the Israeli soldier the Palestinian group seized in June 2006.
"A permanent opening of the [Gaza border] crossings will be linked to solving the issue of Gilad Shalit," an Israeli official quoted Olmert as telling the US envoy.
Mitchell had flown into Israel from Egypt on the second leg of his trip, under instructions from Barack Obama, the US president, to "engage vigorously" in an effort to revive Israeli-Palestinian peace talks.
Mitchell’s arrival in the region coincided with Israeli bombing raids on Gaza early on Wednesday and the killing of an Israeli soldier near the border a day earlier.
Al Jazeera's Barnaby Phillips, reporting from Jerusalem, said: "The original intention, as US president Barack Obama said yesterday, was that Mitchell would go to the Middle East to 'listen and to learn' - to show that the United States is not going to dictate terms in the Middle East.
"Having said that, the situation on the ground here has deteriorated in the past 24 hours.
"Mitchell has had to turn into a little bit more of a firefighter than he originally thought when he scheduled this tour."
Mitchell earlier discussed with Hosni Mubarak, the Egyptian president, an Egyptian initiative aimed at restoring relative calm between Israel and the Palestinians and the re-opening of Gaza's border points.
"The United States is grateful to Egypt for its leadership in bringing about a ceasefire. It is of critical importance that the ceasefire be extended and consolidated," Mitchell said afterwards.
"The United States is committed to vigorously pursuing a lasting peace and stability in the region.
"The decision by President Obama to dispatch me to this region less than one week after his inauguration is clear and tangible evidence of this commitment."
Mitchell is also due to head to Ramallah in the occupied West Bank on Thursday to meet with Mahmoud Abbas, the Palestinian president, who has criticised Israel ahead of the talks.
"Today, we are convinced more than ever, especially after the aggression against Gaza, that Israel does not want peace and we are going to say so to all those who come to see us," he said.
Abbas and Olmert had relaunched peace talks to great fanfare in November 2007 after a six-year hiatus, but the negotiations have made virtually no progress.
Following Israel's war on Gaza, Egypt has been holding separate talks with Israel and Hamas, the Palestinian group, as well as representatives of other Palestinian factions.
Ahmed Abul Gheit, the Egyptian foreign minister, said the talks have "evolved positively" and a "permanent" truce could be agreed in the first week of February.
He said such a ceasefire would lead to the reopening of crossing points into Gaza, where most of the 1.5 million population depend on outside aid, but which has been closed to all but basic humanitarian goods by Israel since Hamas seized power.
Hamas wants the border crossings into Gaza reopened, including the Rafah checkpoint bordering Egypt, to end the Israeli blockade in the territory.
Israel wants to stop the rocket fire and prevent Hamas fighters from using smuggling tunnels under the border with Egypt to rearm themselves with weapons.
"The Israelis' position is extremely tough," Phillips said.
"They are determined to show that the policy of deterrence - which they believed justified the recent attacks on Gaza - worked ... It makes it a very difficult situation for Mr. Mitchell."