"You have proposed a range of dates in the next two months. I will be happy to agree a date that is to the convenience of the inquiry," he wrote in the letter.
Speaking to Al Jazeera, Vincent Moss, the political editor of the Sunday Mirror, a UK newspaper, said: "He'd [Brown] hoped to avoid this until after the election, [there has been] a lot of pressure from the opposition parties, particularly the Liberal Democrats, for him to do it before.
"If he makes a mistake, or is drawn into the planning of this in a way he doesn't like, then it will have an impact on voting day here.
"I think it's a gamble, but I think he must think he's in the clear, otherwise he wouldn't have been so quick to concede, as he did last night, to go before them."
The initial decision to hear from Brown after the elections had drawn criticism from his opponents, who said the prime minister was trying to avoid drawing attention to his role in the war before voters headed to the polls.
Brown served as finance minister when Tony Blair, the then British prime minister, decided to send UK troops to the 2003 invasion of Iraq.
The conflict split British public opinion and spawned a massive protest movement, and much remaining support for the campaign withered away as Iraq descended into chaos.
Brown has since come under criticism for allegedly depriving troops in the field of equipment.
Geoff Hoon, defence secretary during the invasion, said Brown's treasury had failed to allocate enough money to the military build-up in the run-up to the war.
The prime minister's Labour Party is already struggling to cling to office after 13 years in power, with the opposition Conservatives having recently posted double-digit poll leads.