Since Morgan was sacked on Friday following fake pictures of Iraqi prisoner abuse, deputy editor Des Kelly, has been in charge of running the paper on an acting basis.
If Weaver is appointed, the Daily Mirror, both it and rival the Sun, will both be edited by women. The Sun is edited by Rebekah Wade. Insiders have suggested that it would make sense for Weaver to edit the paper, due to its largely female readership.
However, the race to edit the tabloid is not over yet and insiders are also saying claim that Kelly could be in the running as well. Although his management style contrasts unfavourably compared with the Morgan who was popular with staff and known as a charming character.
Weaver is currently in the final weeks of her maternity leave and has worked at The Mirror since 1995. She was named reporter of the year at the British Press Awards in 1994. She then went on to edit M magazine and landed her role at The Sunday title following the resignation of editor Colin Myler who jumped ship to the News of the World.
Other contenders for the role include Phil Hall, former News of the World and Hello! Editor, who is currently editorial development director at Trinity Mirror and Richard Wallace, acting editor of the Sunday Mirror, who has a strong track record at the daily. Other names suggested include The People's editor Mark Thomas and Martin Clark, editor of Ireland on Sunday, who has edited the Daily record and The Scotsman.
Whoever takes over the reins will have the task of living up to one of the most colourful characters in Fleet Street. Morgan is regarded and one of the most influential journalists since the legendary Hugh Cudlip at the title.
Morgan has commanded huge loyalty from his staff. Some reports suggested that some members of staff wanted to stage a Greg Dyke style protest and threaten to walk out over his sacking, but were quickly persuaded to change their minds by more senior members after being reminded that Morgan was in line for a £1m pay-off.
One insider told the BBC: "When he took over eight and a half years ago the Mirror was crap, and he transformed it."
Morgan is currently being tipped for a television career, which is likely to start with an appearance on' Have I Got News For You'.
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