Alonso ended up retiring from the Australian Grand Prix with a suspension problem. His rookie team mate Stoffel Vandoorne lasted the full race distance - and even managed quicker lap times than Alonso.
Webber feels that with little prospect of any improvement at the team, Alonso might call time on McLaren after three years.
"Alonso may not stay with the team," the Australian told the Belgian news agency Sporza. "Maybe Stoffel will have a new teammate soon.
"I can see it happening, that Alonso won't complete the season. He is very frustrated.
"Fernando is not here for sixth or seventh places," the former Formula 1 driver added. "He's not interested in points. He wants to fight for the podium."
Whether Alonso would quit Formula 1 altogether or seek another seat with a top team is unclear.
The two-time world champion was one of the names linked to replace Nico Rosberg at Mercedes. Valtteri Bottas signed a one-year contract with the team in January, leading to suggestions he could be replaced by Alonso or Sebastian Vettel in 2018.
Read the full article on eurosport.co.uk: Webber hints Alonso could be about to quit McLaren