Barca made the decision to extend Valverde's deal for an additional season in February of past year, which preceded an embarrassing Champions League capitulation at the hands of Liverpool just months later, having done similar against Roma in the previous campaign.
Koeman, who scored the winning goal for Barça against Sampdoria in the 1992 European Cup final, has made no secret of his desire to return to the club and has a clause in his contract allowing him to leave for Barcelona after this summer's European Championship.
Setien's appointment may come as a surprise to fans unaware of his previous work, especially given the Camp Nou outfit have been in talks with club legend Xavi about taking the reins in Catalunya. Xavi said he met with Barcelona sports director Eric Abidal on Friday in Doha but did not reveal the nature of the meeting.
The La Liga leaders appear to be close to sacking the Spaniard following their loss to Atletico Madrid in the Spanish Super Cup on Thursday.More news: Ethnic minorities and women mostly miss out while 'Joker' tops Oscars list
"We have to remember that [Pochettino] said that he'd rather be in his farm in Argentina as a farmer than ever becoming Barcelona coach", Montero said.
Setien led Betis to a sixth-place finish in 2018 and qualification for the Europa League but, despite overseeing a thrilling 4-3 victory at Camp Nou early the following season, results tailed off.
However, in a stylistic sense at least, Setien should fit in perfectly at Barcelona having been inspired by Johan Cruyff's side while playing against them in the late 1980s.
Former Barcelona midfielder Xavi Hernandez and Ronald Koeman were both approached to replace Valverde, according to Spanish reports, but neither accepted the role.More news: Talks underway to rescue Flybe
It felt like last season would have been the final straw when Barca blew a 3-0 lead in the Champions League semifinals to eventual champion Liverpool and fell short to Valencia in the Copa del Rey final.
Setien's contract runs through the 2022 season.
Barça sit atop La Liga at the halfway stage, but are ahead of Real Madrid on goal difference alone, and will be without Luis Suarez for the majority of the second half of the season. The Champions League matches against Napoli will be on February 25 and March 18.
His first game in charge will be at home to Granada on Sunday.More news: What's missing from this year's Oscar nominations