While the New Zealander has impressed greatly during his stand-in performances since the Dutch Grand Prix as a replacement for Daniel Ricciardo, it is looking unlikely that it has been enough to secure him a full-time F1 drive.
With Red Bull already eager to hand Ricciardo a seat at AlphaTauri next season, Lawson's only real hope of getting a 2024 drive was for the squad to choose him over incumbent Yuki Tsunoda.
But with it understood that Tsunoda and AlphaTauri have agreed terms to continue into 2024, and an announcement expected to be made at this weekend's Japanese GP, it means Lawson will have to wait for that opportunity to come.
Options elsewhere on the grid look remote, with only a possible Williams seat still available if it does not keep Logan Sargeant.
Instead, the most likely route is for Lawson to take on a test and reserve role at Red Bull as he bides his time until 2025 when there could be a shuffle of the energy drinks company's drivers.
Speaking at the Japanese Grand Prix on Friday, Red Bull team boss Christian Horner suggested it was unlikely that Lawson would be loaned out, but indicated there were plans for him to remain working with Red Bull.
"He's in the Red Bull family," he said. "I think there's only one seat left available, and we've done that in the past when, for example, Carlos [Sainz] went on loan to the previous Renault team.
"But I think if he's not sitting in a grand prix car next year, then he is going to have plenty on his plate with other stuff to do."
Source: Autosport