But the €22 million purchase option included in the original agreement is well beyond what they can afford.
Hamburg head coach Merlin Polzin made the clearest assessment yet of Vieira's impact this season, describing him to the Hamburger Abendblatt as the most influential individual at the club in recent years.
"No player at HSV in recent years has made the difference the way he has," Polzin said. The 25-year-old Portuguese midfielder, on loan from Arsenal since September, finished the campaign with seven goals and six assists in 31 appearances, helping the club retain their Bundesliga status with a 13th-place finish.
Polzin also revealed that Vieira approached him directly.
"He expressed himself warmly about the relationship of trust that existed from the very beginning. He also found personal words — for Loic and for me — that he had on his heart." Polzin added: "I have assured Fábio that we will try everything if a realistically achievable possibility presents itself."
The obstacle is financial. HSV are working from a budget described by Bild as in the high single-digit millions, against a formal option of €22 million. Arsenal are under no obligation to accept less, and the two parties must bridge a gap that Sky Deutschland journalist Florian Plettenberg described as likely to make an agreement difficult.
Vieira himself has not ruled out staying.
"I have to be open to everything," he told reporters. "No matter what the future brings, I was very happy here and will be forever grateful to HSV. The club did everything for me, the support was amazing."
Arsenal are also understood to have agreed a follow-up contract with Vieira at HSV's end, contingent on a transfer fee being settled.
Arsenal paid €35 million to sign Vieira from FC Porto in 2022. He has never established himself as a regular under Mikel Arteta, spending last season on loan at Porto before moving to Hamburg. With Declan Rice, Martín Zubimendi and others now ahead of him in the Arsenal pecking order, a permanent exit makes sense for all parties if the numbers can be made to work.
Whether they can is the question neither club has yet answered.