The Heed are without a manager after the club’s former defender decided it was right to leave the International Stadium.
Gateshead missed out on the National League play-offs on the final day of the season having spent a large majority of the campaign inside the top seven.
Magnay wants a fresh start and the club do not plan on hanging around to find his replacement.
Gateshead last month said the club face "significant challenges" to take the club into next season after a proposed takeover fell through.
Chairman Neil Pinkerton resigned with vice-chairman Bernard McWilliams taking over.
Officials outlined the timeline they are working towards in a statement released on the club website.
It read: It read: “Gateshead FC can confirm that Carl Magnay has tended his resignation as first-team manager.
“We wish to thank Carl for his efforts during his time as manager of Gateshead FC, and wish him the best of luck in his future endeavours.
"We will be looking to appoint a new manager as swiftly and efficiently as possible.
“Applications will close by 5pm on Friday, June 6. Successful applicants will be invited to interview during the week commencing Monday, June 9.”