The 28-year-old has found himself on the fringes of Eddie Howe's squad over the last year, not featuring for the Magpies in the Premier League for the entirety of the 2022-23 term.
Hayden was loaned out to Championship outfit Norwich City as he looked for first-team football, however the midfielder made just nine starts during an injury-riddled time at Carrow Road.
The former Arsenal youngster has featured 171 times for the North-East giants since moving from Hull City in the summer of 2016, scoring seven times over a seven-year period.
According to Football Insider, Hayden is set to undergo a medical at Luton ahead of a switch to Kenilworth Road for the 2023-24 campaign.
The report claims that the move will be a temporary one, meaning that the Englishman would be returning to St James' Park at the conclusion of the campaign, regardless of whether the Hatters retain their Premier League status.
It is understood that Hayden is surplus to requirements at Newcastle, with the 28-year-old failing to appear in any of the club's matches in pre-season or at the beginning of the competitive season.
The former England youth international is said to be training away from Howe's first-team squad as the Magpies look set to offload the player before the window shuts on Friday.
Hayden has three years left on his current terms at St James' Park, although it is now unlikely that the midfielder will be a Newcastle player in 2026.
Luton are in the Premier League for the first time in their history, and Rob Edwards is beginning to target players with top-flight experience in the closing stages of this window.
A host of talent from the Football League has joined the Kenilworth Road club, such as Tahith Chong and Jacob Brown, however the Hatters are hoping to recruit a few players with knowhow of the Premier League.
Former Chelsea man Ross Barkley signed after his spell at French club Nice came to an end, with the playmaker looking to be a standout performer for the top-flight newcomers.
Luton have struggled in their first two league matches, losing 4-1 at Brighton & Hove Albion before suffering a 3-0 defeat at Stamford Bridge against Chelsea last time out.
Kenilworth Road will welcome Gillingham in the second round of the EFL Cup on Tuesday, before the Hatters' unique home makes its debut on the Premier League stage as West Ham United visit on Friday night.