But the three men failed to provide payment details and the vehicle was not picked up, prompting them to use a smaller van from a DIY store instead.
The men drove into pedestrians on the bridge before stabbing people in Borough Market seven days ago.
Police said the men tied 12in (30cm) pink ceramic knives to their wrists and had petrol bombs in the van.
Eight people were killed and dozens more were injured in the attack, which began shortly before 22:00 BST on 3 June.
Khuram Shazad Butt, Rachid Redouane and Youssef Zaghba have been named as the attackers, who were all shot dead.
Scotland Yard wants more witnesses to come forward and is also appealing to companies who hire out vans to report any suspicious activity.
Early on Saturday, police said they had arrested a man, 27, on suspicion of preparing terrorists acts, after carrying out a raid in Ilford, east London, linked to the attack.
In its most detailed description of the attack yet, the Metropolitan Police said the men had also rented a flat in Barking, east London, to use as a safe house.
Inside, they found a copy of the Koran opened at a page describing martyrdom.
They also discovered equipment for making petrol bombs, plastic bottles and duct tape for constructing fake suicide bomb belts, and an ID card for Redouane.
Forensic work at this flat has determined that they acted alone.
The investigation has concluded that 27-year-old Butt was the ringleader. He hired the van used in the attack from a B&Q store in Romford on the morning of the attack.
He is thought to have driven the van, with Redouane and Zaghba in the back, into central London.
At two minutes before 22:00 BST, the van crossed London Bridge heading south. Six minutes later it returned, crossing over the bridge again and making a U-turn at the northern end.
The attack began with the van driving back along the pavements of the bridge, running down three pedestrians before crashing outside the Barrowboy and Banker pub opposite London Bridge station.
The men jumped out clutching the ceramic kitchen knives. Police say the weapons were possibly chosen to prevent them being picked up by metal detectors.
They stabbed five people who had been enjoying the area’s pubs and restaurants.
Police were called within two minutes and arrived eight minutes later, killing the attackers in what officers said was an “unprecedented” volley of 46 bullets.
When briefing reporters, Commander Dean Haydon said the police – from the Met and City of London forces – had shown “incredible bravery”.
He added: “We have stories of people armed with chairs, bottles, anything they could get their hands on with a view to trying to prevent the attackers coming to pubs and bars but also scaring them off to prevent other people being attacked.”
He detailed the actions of four people who put their lives at risk to fight back and help the injured:
* A doctor at a restaurant heard screaming and “with no regard for his own safety” ran outside to pick up an injured man and carry him across the bridge
* A public relations consultant heard the van crashing and gave first aid to a man who’d been stabbed – despite the danger
* A restaurant worker tried to fight off an attacker who came in and stabbed a young woman in the back
* An off-duty police officer was stabbed in the stomach as he tried to disarm one of the attackers
The massive police investigation into the attack continues and by Saturday there had been 19 arrests and 13 buildings had been searched.
So far, 282 witnesses from 19 countries have been questioned but police still want more people who saw what happened to come forward.
In the back of the white Renault van used in the attack officers found wine bottles filled with a flammable liquid with rags tied to their necks. There were blow torches for lighting these “Molotov cocktails”.
The three men had added bags of building gravel and several chairs – possibly to suggest a reason for having hired the van.
Scotland Yard said it wanted to hear from anyone renting vans who might have suspicions about a customer.
Police are also appealing for information about the “distinctive” pink “Ernesto” brand knives the men were carrying.
They said they had not found evidence of anyone else being involved in the plot – or inspiring the attackers to carry it out.
The Met revealed Butt had been arrested for bank fraud in October 2016 but was not charged. He had been opening accounts and closing them again, possibly to launder money.
Police also confirmed there had been a call about him to an anti-terrorism hotline, but no evidence was given that he was planning an attack.
However when he appeared in a Channel 4 documentary, The Jihadis Next Door, last year, officers did view the programme. They decided it was “deeply abhorrent” but not evidence of criminality.
Mr Haydon described the current terror alert as “unprecedented times”. Since March, there have been five planned attacks of which two were prevented and three carried out – in Westminster, Manchester and London Bridge.
“The tempo has increased,” he said, adding that officers at Britain’s intelligence services were “working flat out”.
“Hostile Vehicle Mitigation Barriers” have been added to central London bridges to prevent vehicles being driven onto the pavement.
Meanwhile, police are working with “iconic venues” and music festivals to improve their security over the summer.
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