According to The Sun, Bleakley has agreed a two-year contract with producers, but at lower salary than her previous £100,000.
A source said: "The BBC played serious hardball over this. Their view was that nobody is worth holding on to at any cost anymore.
"They figure there are plenty of stars out there who can replace people who demand too much."
They added that the presenter is happy to work on the programme and is also realistic about the financial pressures in the current economic climate.
Earlier this month, audience ratings for The One Show topped six million in the aftermath of the Carol Thatcher controversy.