Fears that authorities were closing in may help explain the involvement of Laachraoui in the suicide bombing at the airport.
Terrorism experts regard bombmakers, especially those trained in handling sensitive explosives, as among the most valuable and protected members of a terrorist organisation. It is highly unusual for them to participate in suicide attacks themselves.
The computer message also gives apparent insight into the tactics, organisation and motivation of the militants who perpetrated the worst attacks on Belgian soil since World War II, and possibly a deeper look into the wider network linked to last year's Paris massacres.
At least one of the men believed to have been a suicide attacker was deported to Europe from Turkey in July 2015 after Turkey determined he was a militant, Turkish President Recep Tayyip Erdogan said. He suggested that counterterrorism officials had the militant on their radar long before the November Paris attacks or Tuesday's bloodshed in Brussels. Interpol had also issued a "red notice," effectively an international arrest warrant, for one of the suspects at the request of Belgian authorities.
In the note - discovered on a computer dumped near an apartment containing bombmaking material - one of the suspected suicide attackers, Ibrahim El Bakraoui, apparently described feeling pressure bearing down.
He wrote that he was "in a hurry, no longer know what to do, being searched for everywhere, no longer secure," according to Van Leeuw's description of the message, which was not made public.
Authorities now believe that the bombers had close connections to the Paris attackers.
The same bombmaker may have been involved in both attacks, and Khalid El Bakraoui is believed to have used an assumed name to rent a Brussels area apartment where Paris suspect Salah Abdeslam's fingerprints were found last week.
The computer file does not mention Abdeslam by name, but it says the attackers feared that if they did not strike quickly, they risked winding up in prison alongside "him".
"If they drag on, they risk finishing next to him in a cell," Van Leeuw said, paraphrasing the content of the file.
Van Leeuw described the file as a "will" discovered on a computer. He did not explain why authorities believed the computer belonged to Ibrahim El Bakraoui.
The latest violence has left European leaders again scrambling for ways to plug holes in security, even though it became increasingly clear that at least Ibrahim El Bakraoui had repeatedly passed through security nets without being detained by European authorities.
In Brussels, leaders called for new powers to fight terrorism, although it was unclear whether there would be any progress this time, since similar proposals were made, then rejected, after last year's attacks in Paris.
French Prime Minister Manuel Valls repeated calls for sweeping new powers to be given to European intelligence agencies, warning that the future of European unity is at stake.
"If the European project is running out of steam, if the populists are gaining in popularity, it's because a lot of speeches are not followed up in reality," Valls said in Brussels, criticising the vows for reform that have followed other recent terrorist attacks but yielded few concrete changes.
"In the years to come, the [EU] member states will have to invest massively in their security systems," he said.
"This is a kind of scenario every capital in Europe feared since the November attacks last year. A mixture of foreign fighters coming back with experience, local sympathizers on the other hand," said Rik Coolsaet, a terrorism expert at Ghent University who has advised the Belgian Government on how to fight radicalisation. "You have such a large number of soft targets, and you cannot secure all of them."