The 25-year-old Sweden international headed back to Tyneside over the weekend after spending time working on his fitness at former club Real Sociedad, in Spain.
English Premier League champions Liverpool have made a £110million (A$226m) bid, which was swiftly rejected by the Magpies.
And if he does leave, it will almost certainly be for a British record fee.
Isak reported to the club's training ground on Monday morning as the rest of the squad returned from their pre-season trip to Singapore and South Korea.
He missed the tour having cited a thigh injury amid fevered speculation over his future.
Newcastle have repeatedly insisted they have no intention of selling their star striker - whom they value at £150m (A$308m) - this year.
But his signal that he wanted to explore the possibility of a move away from St James' Park, where he still has three years remaining on his contract, brought matters to a head.
Howe, who desperately wants to keep the player, has been diplomatic about the situation in public but he is unlikely to tolerate disruption of the team ethic.
Indeed, speaking to reporters before leaving Seoul after Sunday's 1-1 friendly draw with Tottenham, he said: "You have to earn the right to train with us.
"We are Newcastle United. The player has a responsibility here to be part of a team and part of a squad - you have to act in the right way."
Newcastle have made a bid for RB Leipzig's Benjamin Sesko - also a target for Manchester United - and are interested in Brentford's Yoane Wissa.
Meanwhile, Liverpool have sold winger Luis Diaz to Bayern Munich and could be about to sell striker Darwin Nunez, who is attracting interest from Saudi side Al-Hilal, further boosting their potential spending power.
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