"I have apologised to the Prime Minister and offered my resignation as Transport Minister.
"She has declined my offer but chosen to transfer my responsibility for the Civil Aviation Authority to Associate Transport Minister Julie Anne Genter.
"I have referred the matter to the Civil Aviation Authority who will follow whatever processes they deem appropriate," Twyford said.
National MP Gerry Brownlee was fined $2000 for breaching airport rules when he bypassed security to board a domestic flight in 2014 while he was Transport Minister.
Brownlee and two of his aides breached security at Christchurch Airport as they were running late for a flight.
In his letter to the CAA director, Twyford said he wanted to draw his attention to the incident on May 17.
"I believe this was in breach of the Civil Aviation Rules regarding the use of electronic devices on aeroplanes.
"I apologise for this breach and understand that you will take whatever action you deem appropriate," he said in the letter.
Speaking to reporters later, Twyford said he made the call from a plane he had boarded in Wellington.
"That's unacceptable. It's particularly inappropriate for a Minister of Transport to do that.
"Today I informed the Prime Minister of this matter and offered my resignation.
"I made a mistake and I clearly wasn't thinking straight at the time. I recognise that. It was unacceptable and I apologise unreservedly."
Twyford said he was making a call to a staff member and thought it had been important at the time.
"In hindsight it doesn't excuse or justify breaking the rules."
Twyford said he had not given the matter a "moment's thought" until today, when he received a written parliamentary question about it from Judith Collins.
About tendering his resignation he said: "That's not something that any minister wants to do."
The Prime Minister had expressed her disappointment before a decision was made to transfer his CAA reponsibilities.
Ardern said she expected all her ministers to act in accordance with the rules.
"As Transport Minister it is even more important that Phil abides by civil aviation laws," she said in a statement.
"It isn't appropriate for him to have responsibility for the Civil Aviation Authority in the event that it investigates this incident, so it's appropriate to transfer those responsibilities to Julie Anne Genter."
The CAA said the Director of Civil Aviation had received details of the alleged breach of Civil Aviation Rules by Twyford and would examine the facts before determining what action, if any may follow.
"It is premature to speculate on what might happen next. The CAA won't comment further while inquiring into the matter," a spokesman said.
Collins told the Herald she had been alerted to the incident by a member of the public.
"It is important that as the Minister in charge of Civil Aviation that he comply with the rules that everyone else has to.
"Gerry Brownlee had to resign as Minister of Transport when he was found in breach of the rules and it's important that there are standards."
In 1991, when Minister of Police, John Banks was fined $750 for answering his early-model mobile phone on a flight.
It is a second black mark against Twyford within a week.
He was told off by Ardern after he called Treasury officials "kids fresh out of university" when asked why their figures on construction investment differed to his.
