Mr Corbyn made the nominations before he stepped down as Labour leader, but it is understood that the party has been informed by the House of Lords Appointments Commission, which vets candidates for propriety, that they were “not agreed”.
Mr Watson is believed to have been rejected over his support for allegations of a paedophile ring at Westminster made by the fantasist Carl Beech, who was later jailed. Harvey Proctor, a former Conservative MP smeared by the allegations, has led a campaign to deny Mr Watson a peerage.
A source told The Independent that, unlike the other nominations, there was “more of an argument” about Mr Watson’s case, and “a lot of umming and ahhing”.
It is understood that Mr Bercow’s nomination was rejected because the commission felt Mr Corbyn was not entitled to nominate him because the former Conservative MP is “not Labour”.
His nomination would have been expected to come from the government, which has traditionally nominated retiring speakers to the upper house. But the prime minister failed to recommend him, after Conservatives accused the former speaker of abusing his position as an impartial chair to campaign against Brexit.
The reasons for rejecting Ms Murphy’s nomination are not known, although a source described her case as a “slam dunk”, and anti-racist groups including Hope Not Hate had lobbied against her elevation to the peerage while the Equality and Human Rights Commission is still carrying out its investigation into antisemitism in Labour ranks.
A spokesperson for the House of Lords Appointments Commission refused to comment on individual cases, but explained that its role is “to advise the prime minister if it has any concerns about the propriety of a nominee”.
The commission defines “propriety” to mean: “the individual should be in good standing in the community in general and with the public regulatory authorities in particular; and the past conduct of the nominee would not reasonably be regarded as bringing the House of Lords into disrepute”.
However, in Mr Bercow’s case it is possible that his nomination has been refused on procedural grounds, rather than on the basis of allegations of bullying made against him, an investigation of which was never completed.
A spokesperson for the Labour Party refused to comment, but party sources were keen to distance Sir Keir Starmer from the nominations, which were made on behalf of the party.
“These were appointments made by Jeremy Corbyn; we are not involved in the decision-making process.”
All political nominations for the dissolution honours list after the last election, including as many as 20 Conservative peers, have been delayed until at least the autumn by the coronavirus crisis.
Mr Bercow and Mr Watson have not responded to requests for comment.
Ms Murphy said she did not know that her nomination had been rejected. She told The Independent: “I speak to Jeremy every other day.”
However, The Sunday Times reported that Mr Corbyn had been informed about the commission’s decisions and was offered the chance to make substitute nominations.
He is reported to have initially agreed to nominate other candidates, but to have changed his mind.