WBC cruiserweight champion Bellew, 34, produced an upset to stop David Haye in a heavyweight contest last month.
But when asked if he would fight WBA and IBF champion Joshua, Bellew said: "Absolutely not. I don't need that kind of fight. He's far too good for me."
His promoter Eddie Hearn said he could fight WBO champion Joseph Parker next.
New Zealand's Parker, 25, will defend his title against Romanian Razvan Cojanu in Auckland next weekend.
Speaking after Joshua's 11th-round stoppage of Wladimir Klitschko, Bellew said: "I'm still dreaming about winning the heavyweight world title but a different portion to the one AJ has."
The Liverpudlian said he was considering retirement following his surprise victory over Haye last month on his heavyweight debut.
But on fighting Parker, he added: "I say Eddie knows what he's doing."
Promoter Hearn confirmed that an all-British Bellew-Joshua fight was not on the cards.
"Tony and Joshua are good friends," he told BBC Radio 5 live's Sportsweek programme on Sunday morning.
"Tony Bellew has got his eye on some of the smaller guys. Fighters like Joshua are giants and Bellew isn't, but he's in the mix we talk about for challenging for world heavyweight titles."
Bellew is not a natural heavyweight, having stepped up from the cruiserweight division to fight David Haye in March.
He weighed in at 96.6kg for that fight, while Joshua was at a career heaviest of 113 kg against Klitschko on Saturday - a difference of almost 20kg.
But also in the mix for Bellew is WBC heavyweight champion Deontay Wilder - and the American wants a unification fight against Joshua, should he take the WBO belt off Parker.
"My plans are to get the WBO belt and once I get that, if I'm given a mandatory challenger, get that out of the way and set up a mega fight early in 2018," said Wilder.

Lara Clarke, from Brownhills, has previously made near-life-sized cakes of Hollywood stars Johnny Depp and Jennifer Lawrence.
Her latest creation, baked to celebrate the Defender model, fed 2,000 people at an event at the factory in Solihull on Monday.
The half-scale replica weighed 600kg and featured working headlights.
Comprising 32 sponge cakes, 50kg (110 lbs) of butter cream, 20kg (44 lbs) of ganache and 50kg icing, it took her 150 hours to produce.
Ms Clarke said the bake was so heavy it had to be delivered by fork lift truck.
"We arrived by van and everyone was coming round to have a look," she said.
"They couldn't believe it was a cake. There was plenty to go round and everyone on the factory floor got a piece."
It was baked to celebrate the two millionth Defender rolling off the production line - the figure includes its predecessors the Series I, II and III models.
Ms Clarke said it started after she tagged JLR in a tweet about a small Land Rover birthday cake.
"They got in touch and asked if I could make a cake in the shape of an engine to mark the opening of their new engine plant.
"A woman then asked me if I could make a Land Rover cake.
"It was originally meant to be quite small, but I asked them if they wanted something that would really impress people."
Ms Clarke said she was getting married in October, a week before the Cake International competition at Birmingham's NEC.
Rather than enter this year, she said she planned to bake "the best wedding cake that's ever been seen".
Tony Bellew still retains hope of winning a world heavyweight title but says he will not fight fellow Briton Anthony Joshua.
[NEXT_CONCEPT]
A giant cake has been made in the shape of a Land Rover by an award-winning amateur baker from the West Midlands.