Two other junior ministers are also reported to be resigning.
Revelations over the expenses of members of parliament in the last month have led to a string of resignations and suspensions from all major political parties and sparked outrage among taxpayers.
Blears was criticised for making tens of thousands of dollars tax free by selling a home she was using public money to maintain. She later repaid $21,500 of the money.
The latest developments have put significant pressure on Brown, who many observers believe has lost control of his cabinet.
Nick Clegg, leader of the opposition Liberal Democrats, has said that the government is in its "death throes".
The Guardian newspaper, which is traditionally sympathetic to Brown and Labour, his ruling party, has also turned against the leader, saying it is "time to cut him loose" in an editorial published on Wednesday.
It said Labour "faces its worst defeat in its history tomorrow but the prime minister does not recognise his direct responsibility for the mayhem".
Thursday's elections are expected to show a strong swing against Labour, with an Ipsos Mori poll involving 1,000 people giving the main opposition Conservative party a 22-point lead over Labour.
A Sunday Telegraph/ICM poll at the weekend suggested Labour would come in third in the European elections with just 17 per cent of the vote, behind the Conservatives and Liberal Democrats.