Six weeks since a fragile ceasefire took effect, talks to end the war have made little progress, while soaring oil prices have raised concern over inflation and the impact on the global economy.
Trump also faces domestic pressure before November's midterm elections, with his approval rating dropping close to its lowest since he returned to the White House on the surge in fuel prices.
Pakistan Army Chief Asim Munir would decide on Thursday whether to travel to Tehran as part of the mediation effort, three sources familiar with the negotiations told Reuters.
They sought anonymity because they were not authorised to speak to media.
Pakistan's interior minister was in Tehran on Wednesday.
"We're speaking to all the various groups in Iran to streamline communication and so things pick up pace," one of the sources said.
"Trump's patience running thin is a concern, but we're working on the pace at which messages are relayed from each side."
Earlier, Iran's ISNA news agency said Munir would travel to Tehran on Thursday for consultations.
"Believe me, if we don't get the right answers, it goes very quickly. We're all ready to go," Trump told reporters at Joint Base Andrews.
Asked how long he would wait, Trump said: "It could be a few days, but it could go very quickly."
Trump reiterated his determination not to allow Iran to acquire a nuclear weapon.
"We're in the final stages of Iran. We'll see what happens. Either have a deal or we're going to do some things that are a little bit nasty, but hopefully that won't happen," Trump told reporters earlier in the day.
Iran's Revolutionary Guards has warned against renewed attacks.
"If aggression against Iran is repeated, the promised regional war will extend beyond the region this time," it said in a statement.
Iran submitted its latest offer to the US on Monday.
Tehran's descriptions suggest it largely repeats terms Trump previously rejected, including demands for control of the Strait of Hormuz, compensation for war damage, lifting of sanctions, release of frozen assets and the withdrawal of US troops.
The Strait of Hormuz, which carried a fifth of global shipments of oil and liquefied natural gas before the war, has been all but closed since it began, causing the most serious disruption to global energy supplies in history.
On Wednesday, Iran released a map showing a "controlled maritime zone" at the strait and said transit would require authorisation from an authority set up to control the area.
It said it aimed to reopen the strait to friendly countries that abide by its terms.
That could potentially include fees for access, which Washington says would be unacceptable.
Two Chinese supertankers carrying a total of about four million barrels of oil exited the strait on Wednesday, while a South Korean tanker with two million barrels of crude loaded in Kuwait was also crossing the strait in co-operation with Iran.
US-Israeli bombing killed thousands of people in Iran before the ceasefire.
Israel has also killed thousands more and driven hundreds of thousands from their homes in Lebanon, which it invaded in pursuit of the Iran-backed Hezbollah armed group.
Iranian strikes on Israel and neighbouring Gulf states have killed dozens of people.
