McLaren confirmed on Friday the 21-year-old Australian will race for them after the sport's Contract Recognition Board ruled in their favour against Renault-owned Alpine, who had also wanted him in their car.
Piastri is not attending this weekend's Dutch Grand Prix at Zandvoort, remaining at Alpine's factory at Enstone in England to carry out simulator work.
He already has a role with both teams after McLaren signed a reserve sharing agreement with Alpine in March, before the start of the season.
"He'll continue this weekend. There's only a couple of days left, today and tomorrow. And then on Monday we'll get together with him and decide how we go forward," Szafnauer told reporters at Zandvoort.
"We'll decide on Monday," he repeated when asked whether Alpine would consider an early release to McLaren. "I don't want to think about it here because there's more considerations to be had."
The Italian Grand Prix is next weekend at Monza, a circuit where Piastri had originally been expected to take part in the first free practice with Alpine as part of a programme preparing him for Formula One.
There is now no benefit for Alpine in keeping Piastri embedded with a team fighting McLaren for fourth in the championship.
Szafnauer said Alpine had plenty of drivers interested in replacing double world champion Fernando Alonso for next season, now that Piastri was not an option, and would take time to consider.
"You want the fastest driver that you can get, one with experience, consistency and speed. We'll be looking at who's available to fill that role," he added.
The frontrunner in paddock speculation is Frenchman Pierre Gasly, who would have to be freed from a contract with Red Bull-owned AlphaTauri.
Other options would be Australian Daniel Ricciardo, whose place Piastri is taking, and Haas's German driver Mick Schumacher.
Piastri says 'breakdown in trust' triggered move
Piastri said a lack of clarity and a 'breakdown in trust' with Renault-owned Alpine played a part in his decision to join Formula One rivals McLaren.
McLaren confirmed at the Dutch Grand Prix at Zandvoort on Friday, after winning a contract dispute, that the 21-year-old Alpine reserve driver would race for them next season.
"I felt McLaren was a great opportunity. They were very straightforward and very keen and enthusiastic to have me," he told Formula1.com in an interview published on Saturday.
"To be completely honest, there was a lack of clarity around my future at the team at Alpine."
Piastri won the Formula Two title last year, after winning Formula Three the season before that, but his progression stalled with no seat available in Formula One.
He has spent this year embedded at Alpine, who had drawn up an intense preparation programme, but the future was uncertain with the team discussing a contract extension with double world champion Alonso.
Alonso, who had wanted a two-year deal, last month announced his departure for Aston Martin at the end of the season. But Piastri had already signed for McLaren by then, inking a contract on July 4.
"The lack of clarity around my future, and ultimately a breakdown in trust, I felt the very attractive offer of McLaren and the positive dealings with them thus far were all reasons why I felt McLaren was where I was best off," said Piastri.
Alpine announced in a statement in August that Piastri would be racing for them but the Australian then posted a message on social media ruling that out.
He told the website he was disappointed with Alpine's handling of the situation and said his management had twice informed the team previously that he would be leaving.
"It was quite upsetting as the announcement was false and it also denied me the opportunity to properly say goodbye to everyone at Enstone," he said.
Alpine principal Otmar Szafnauer said the team had delivered everything promised in a heads of terms agreement and had shown loyalty to the youngster.
