Border Force staff can be redeployed to help deal with a backlog of 30,000 passport applications without compromising security or leading to huge queues at ports and airports, officials have insisted.
It has urged employees to volunteer for up to six weeks to join efforts to process tens of thousands of applications, an internal email seen by the Daily Telegraph revealed.
Commons home affairs committee chairman Keith Vaz said shifting frontline staff would be "wholly unacceptable" and shadow home secretary Yvette Cooper said it raised serious questions.
But the Border Force said temporary transfers could be made with "no impact" on security or other operations.
"Border Force is considering how it can provide short-term support to HM Passport (Office) without any negative impact on their services," a spokesman said.
"Security is our priority. This is a short-term arrangement that will have no impact on Border Force operations or on our ability to protect the border"
It comes amid mounting criticism that passport service has descended into chaos, with many families fearing they might have to cancel their summer holiday plans because of long delays.
The email, the newspaper reported, was signed by the Force's director of national operations Philip Astle and sought "volunteers to provide assistance" to colleagues at the Passport Office.
"I know we are entering Border Force's busiest time of year, with a number of high-profile events also on the horizon," he noted.
"Many of you will be an integral part of Border Force's work to keep the border secure, and this valuable role will continue.
"However, for those of you who are willing and able to provide some support to HMPO, I would ask for your swift response to your director's request for assistance.
"I know that as dedicated and committed officers of Border Force, I can rely on you to provide support where it is needed."
Quoting sources within the agency, the newspaper said staff had been offered "as much overtime as they can handle", hotel accommodation and generous allowances.
Labour MP Mr Vaz said: "It would be wholly wrong to transfer any frontline officers from the important work they do in the Border Force, where they are securing the borders of this country, to an office just to clear a backlog.
"We do not want a repetition of the scenes that we had at our ports because staff have been redeployed.
"This attractive offer of unlimited overtime will mean staff are not concentrating on the important job they have, which is to secure the borders.
"Rather than redeploying staff from critical areas they should be seeking to recruit new staff and give people a realistic timetable for dealing with this."
Ms Cooper said the Home Secretary, who was forced by the Opposition to account for the backlog in the Commons and promised extra staff among other solutions, was " lurching from one crisis to another".
"Theresa May's desperate decision to move staff off border security to the Passport Office for six weeks raises serious questions," she said.
"What implications will this have for security and queues at our borders, especially after the bordersgate fiasco a few years ago? We already know that borders staff were complaining about shifts not being filled at Heathrow before these further reductions.
"Does the Home Secretary anticipate it taking six weeks for the backlog at the Passport Office to be sorted?
"How many more people will have to cancel holidays, miss weddings and sit out of business meetings in this time?
"The Home Secretary must today guarantee that removing staff from the border agency does not leave our borders unprotected, that unfilled shifts will be filled and that ministers will act on warnings from officials.
"Does the Home Secretary have any kind of grip of what is going on at all?"