Alonso has won 32 races over his Formula 1 career
Team boss Otmar Szafnauer said Alpine offered a deal for 2023 with an option for 2024, but 41-year-old Alonso "wanted something longer".
Szafnauer said Alpine's concerns were that Alonso's abilities may wane as he progresses further into his 40s.
Australian Oscar Piastri has been named as Alonso's replacement for 2023.
However, Piastri has since contradicted that claim and says that no contract has been signed and he will not be driving for the team.
McLaren have also had talks with Piastri, Alpine's reserve and the reigning Formula 2 champion.
Szafnauer said Alonso had assured him upon leaving the Hungarian Grand Prix on Sunday night that they would continue their talks in the coming days, only for Aston Martin's news release to arrive on Monday morning.
"We offered him a one-plus-one deal," Szafnauer said. "We discussed with Fernando, 'Look, if next year at this time you're performing at the same level, of course we'll take you.'
"But he wanted more certainty: 'Independent of performance, I want to stay for longer.'
"That was the crux of going one-plus-one as opposed to two-plus-one or three-plus-one or three years.
"There does come a time when something happens physiologically and you don't have the same capabilities as when you were younger.
"It happened to Michael Schumacher at 42 - he was not the same driver as at 32 as 35. It happens in cricket, which is a far less physical sport. And it happens to racing drivers, too.
"So we were in favour of yes, if you're performing at a high level we'll keep you. But let's do it one year at a time."
Szafnauer said he was aware of Aston Martin's interest in Alonso following Sebastian Vettel's decision to retire at the end of the season - and had seen Alonso and team owner Lawrence Stroll leaving the Aston Martin motorhome together in Hungary.
"I was confident even with the discussions - there is nothing wrong with exploring - that we were very close (to a deal)," Szafnauer said.
"I did ask the question, but I was told 'no I haven't signed anything'. So I was a bit surprised."
Szafnauer said he had exchanged messages with Alonso but had not spoken to him since, because he was on a boat in the Greek islands. But he was confident that the relationship between driver and team would be strong enough to produce good results for the remainder of the season.
"I have absolutely no worries about the rest of the season," Szafnauer said. "Our goals are clear We want to finish at least fourth in the constructors'. Fernando realises that, too.
"I have texted him. I will call him. He is a competitor and a professional and once he puts the helmet on he wants to win. Nothing else is on his mind in the car other than finishing as high as he can."
Szafnauer, who left Aston Martin at the end of last year before joining Alpine, said he was "confident we can out-do the team Fernando is going to during the time he is there".
-- Courtesy of BBC Sport