Bajcetic arrived as a centre-back from Celta Vigo's youth team but soon started to forge ahead as a defensive midfielder. He signed a new contract in the 2022/23 pre-season after impressing the Liverpool coaching team in the first-team's warm up games. “He has amazing physical characteristics. He is extremely quick, makes good recoveries, has a great spring, is really impressive in the air and carries the ball cleanly out of defence,” Alex Otero, youth football co-ordinator at Celta, told The Athletic in January 2021. “It was clear he was a standout player in his age group. He can really play and help construct attacks but he is also noticeable for his comfort with his left and right foot and the ability to play on the right or left of the centre-backs. That is quite rare."

Still two months short of his 18th birthday, Bajcetic made his first team debut on 27 August 2022, coming off the bench to play the last twenty minutes of a 9-0 win over Bournemouth. He made three substitute appearances in the Champions League group stages and scored his first goal for the club when he came off the bench in a 3-1 win over Aston Villa on Boxing Day. Bajcetic's rise at Liverpool has been rapid in the troubled 2022/23 season and he was rewarded with a new contract in January 2023. Throughout 2023/24 Bajcetic struggled with a calf injury and was limited to just three appearances, two of them in the Europa League. His one Premier League appearance was as a 65th minute substitute at home to Tottenham on 5th May. Bajcetic's lack of sharpness showed and the Reds won 4-2, it had been 4-0 when he came on. Jurgen Klopp admitting is was a mistake to rush the promising teenager back, saying: "He comes on and it’s like you arrive on the motorway with your bicycle."

Bajcetic was loaned to Red Bull Salzburg for the 2024/25 season, where Pep Lijnders, his former coach at Liverpool is manager. Arne Slot sees a big future ahead of the youngster: “I hope it will be a big year for him because he’s a very talented player and I think he could become a very important player for us, but only if he gets playing time now,” said the Liverpool head coach. “If you are that young and you’ve been out for so long, the only thing you need is playing time. He could have playing time over here, but we’ve got six or seven very good midfielders, so his playing time here will always be more limited than we expect him to play at Salzburg. So for him, and us, we think it’s best for him to play at the moment, instead of maybe playing a few times and then being on the bench for a long time again. It’s very difficult to keep a player fit, who’s been out for 15 months when you only play him once in a while. Things didn't work out as planned, with Lijnders being sacked halfway through the season as . After making eight starts and eleven substitute appearances, he was recalled in January and loaned to Las Palmas, who were battling relegation from La Liga. Although he was unable to save the Canary Islanders from the drop, he got far more game time and was praised for some battling performances.