Commonwealth governments have rejected a fresh attempt to give their secretary-general, Baroness Scotland, an automatic second term.
In a letter seen by the BBC, Boris Johnson says a “significant and diverse number” had opposed the idea.
He suggested a decision should be taken at their next meeting, which has been postponed amid the pandemic.
An extension of the Labour peer’s contract will be made until then, unless there were objections, he said.
The 26th Commonwealth Heads of Government Meeting had been due to take place in Rwanda and would have brought together leaders from the 54 Commonwealth nations.
At the meeting, the leaders were due to appoint their secretary-general for the next four years.
Baroness Scotland’s term of office had been due to end in March, but with the meeting planned for June, it was agreed that she would get a three-month extension to her contact.
With the postponement of the June meeting, Barbados proposed that the Labour peer should be reappointed automatically without facing re-election.
But Mr Johnson – who is Commonwealth chair-in-office – said in response to an earlier letter detailing the suggestion from Barbados, “a significant and diverse number of colleagues from across the Commonwealth have communicated that they do not support that proposal”.
He said they “prefer that the decision on the next substantive secretary-general appointment should be taken at CHOGM (Commonwealth Heads of Government Meeting), according to our established practice”.
“Therefore, I must respectfully confirm that there is no consensus for the immediate reappointment of the secretary-general for a second four-year term.”
He went on to propose that Lady Scotland’s current contract is extended until the leaders meet.
A Commonwealth spokesman said: “A member state wrote to the chair-in-office to request that heads of government consider reappointment now due to the uncertainty of when CHOGM (Commonwealth Heads of Government Meeting) will be able to take place due to the unprecedented Covid-19 pandemic.
“The chair-in-office wrote to member states on 8 June 2020 to state that consensus was not reached for reappointment to occur now in the absence of CHOGM.”
A UK government spokeswoman said: “We don’t comment on private correspondence between Commonwealth member states.
“Decisions on the appointment of secretaries-general are a matter for heads of government. In the past, those decisions have been made at CHOGMs.”
She added the UK government will work with other Commonwealth countries on the process of appointing the next secretary-general.