Around 1,000 people have signed so-called "satisfaction notices" so far this year, which is twice the figure for the whole of 2007, Sveriges Radio reports.
Morud Habied from Uzbekistan said his case coordinator at the board gave him no option but to sign.
"She said that even if I appealed their decision, the court would go down the same route," he told Sveriges Radio.
Habied's legal representative, Mattias Skarerlius, said the pressure from the board to sign satisfaction notices had intensified during the autumn.
He said people who were told they could not stay in the country were informed "in a very aggressive manner that it is easier for the them go home on their own and there is simply no use in appealing".
But the Migration Board's legal chief Mikael Ribbenvik insisted each individual was entitled to decide how to proceed with their case.
"There's no incentive to put pressure on anyone," he told Sveriges Radio.