Dyche, the current boss of Burnley is the betting favourite and his odds shortened all throughout Monday afternoon when the Daily Mail reported he was the likely choice to replace Koeman.
Former Borussia Dortmund boss Tuchel seems to be the fans choice to replace Koeman and according to a report from the Daily Mirror late on Monday night, seems to be the choice of Everton majority shareholder Farhad Moshiri.
Bill Kenwright and Everton chief executive Robert Elstone met with Koeman after he took Monday morning’s training session to deliver the news that he had been relieved of his duties as Everton boss.
Moshiri made the decision on Monday despite saying only recently he had full belief in Koeman to turn the extremely poor early season start around in the coming weeks.
Being on hand at Goodison Park to see Arsenal tear Koeman’s side part had to have quickened the process in letting the 54-year-old move on from his role as manager.
Needs to be a quick turnaround
Sacking Koeman relatively early into the Premier League campaign means that whoever his successor is, has plenty of time to turn the season around.
The Blues will want to quickly install a new permanent boss as to maximise contact time with the squad to allow for evaluation ahead of possible spending in the January transfer window.
David Unsworth is likely to take temporary charge whilst Moshiri and Kenwright conduct their search for a new boss and it remains to be seen if he’s going to be a runner in the race for the role.
The Everton under-23 boss would prove a popular choice with the Goodison Park faithful but again, Moshiri is looking for a big name to fill the vacancy.
Tuchel, being out of a job, cuts out the need to negotiate a settlement payment with a club to speak to him and then release him from his contract but there are reports he sees himself at a higher level, possibly waiting for a Champions League job to become available.
To free up Dyche from his role at Burnley, the Blues would need to negotiate a settlement payment with the Clarets. That can take time and with the Lancashire side possibly unlikely to let their boss leave quite so easily, it could be a time-consuming process for the Everton decision makers.
Other possible choices
Moshiri and Kenwright will have their key targets but will want to assess all possible options to them before definitively deciding on the man to take Everton forward.
The Blues are also reportedly interested in making contact with Watford for Portuguese boss Marco Silva.
The 40-year-old may not be the first choice of the Goodison Park brass and whilst he isn’t likely to budge from his role at Vicarage Road any time soon, he isn’t a half bad consolation prize if it comes to it.
Contact is also expected to be made with Carlo Ancelotti but the Italian veteran manager remains a dark horse in the Blues search for their third manager in as many years.