Bowyer took the Addicks up from League One last season but has entered the final month of his Charlton contract and could be available this summer.
The Baggies interviewed former West Ham boss Bilic last night, and he is the current front-runner for the position.
But Albion’s hierarchy also want to speak to Bowyer after an impressive first season in management.
Charlton are keen to re-open negotiations with him over a new deal and Derby County are also reportedly interested in Bowyer should current boss Frank Lampard depart this summer.
The 42-year-old former Newcastle midfielder is just the latest name to be added to Albion’s managerial search.
He joins Bilic on a list of candidates that also includes former Brighton boss Chris Hughton and current Sheffield United manager Chris Wilder.
Former Wolfsburg boss Bruno Labbadia has fallen down the pecking order and is now unlikely to be appointed.
Quique Sanchez Flores emerged as another potential candidate, and worked with technical director Luke Dowling at Watford, but the 54-year-old is currently employed by Shanghai Shenua and may be difficult to entice away from China.
The news Bilic had been interviewed was well-received by fans, who have backed the Baggies’ move for him on the Express & Star website.
Seven out of 10 readers voted in favour of appointing the Croatian this summer.
Bilic guided West Ham to a seventh place finish in the Premier League in his first season in England, earning them qualification to the Europa League.
West Ham finished 11th the season afterwards before a poor start to the 2017/18 season cost him his job.
After almost a year out of management, Bilic took over Al-Ittihad but only lasted five months.
Bowyer has only been a manager since March 2018, but he caught the eye during his first full season at The Valley.
Hughton, who was sacked by Brighton at the end of last season, is still in the running for the job.
It's believed he would need to take a pay cut from his Premier League salary to join the Baggies.
And Albion are still keeping tabs on Wilder’s situation at Sheffield United, where an ongoing legal battle between owners Kevin McCabe and Prince Abdullah of Saudi Arabia is causing tension.