The reduction of the 32-year-old's presenting role follows the announcement last month that her One Show colleague Adrian Chiles will only front the ITV show from Monday to Thursday.
Daybreak's sports editor Dan Lobb will take over Chiles's position on the final weekday edition of the breakfast programme.
"This is symptomatic of just how difficult things are on Daybreak at the moment," a show source told the Daily Mail. "We need to recapture the warmth of GMTV and viewers adore Kate, especially when she is teamed with Dan. They haven't taken to Adrian so well and they are only just starting to warm to Christine."
Garraway and Lobb are due to take their spots on the sofa from September following the rugby World Cup. Several editions of the troubled show are expected to be axed during the tournament for notable early morning matches.
Daybreak bosses have claimed that Chiles and Bleakley agreed to have their roles reduced as part of their contracts.
Daybreak has continued to struggle with poor ratings of around 700,000 viewers, less than half of the 1.5m collected by its rival BBC Breakfast and 200,000 less than its predecessor GMTV achieved before its axing in September.