The Hungary international got his first appearance of the season for the Black Cats when they ended the campaign with a 1-0 home loss to Arsenal.
Fulop showed his talent with a couple of fine saves, but wants to be a number one and has conceded that it will be tough to steal the jersey from Craig Gordon.
Boss Roy Keane would love to keep the former Spurs goalkeeper but has accepted his reasons for wanting to go.
Fulop said: "The manager accepts that I have a desire to play week-in, week-out.
"It was great to play against Arsenal – I enjoyed it a lot and I think I did my job. But I want to play in these big games every week.
"I am in my national team and if I want to keep my place I need to play every week.
"Sunderland is a fantastic club and it is great to be here. I have no regrets about coming, but if I want to further my career, I need to play regular football.
"Craig Gordon has done well this season and I think he will be number one here for a lot of years."
Fulop joined Sunderland on an emergency loan in November 2006 and made the move permanent in January last year for a £500,000 fee.
But Keane preferred in-form Darren Ward for most of last season in the Championship and Fulop featured only five times.
Gordon's arrival for a club record £9million made it even tougher for the 25-year-old, who played 27 games on loan at Leicester in the first-half of the season before being recalled.
With Keane already saying publicly that he expected to lose the 6ft 5in stopper, Fulop didn't expect to feature while Gordon was fit.
"I was a bit surprised to play but I have be preparing for every game in case Craig got a knock," he said. "This is a massive club and you have to fight for every minute you get on the pitch.
"I am sure only good things are coming for Sunderland and playing in front of more than 40,000 great fans is exciting, but I need to be somewhere I can play every week."
Fulop has been named in Hungary's squad for games against Greece and Croatia later this month.
Keane said: "I know he's keen to play first-team football and I understand that, players get impatient. But if he is to leave – and it's a big if – we'd have to get the right offer
"I thought Marton did well against Arsenal and it just goes to show what a top keeper he is. He's playing for his national team and he's been instructed by the manager there to get first-team football, which I understand – we'll see what happens."