Marcelo Bielsa has arrived in London for further talks with Everton's hierarchy as they search for a successor to Frank Lampard.
The 67-year-old former Leeds United manager is a favoured choice of Everton owner Farhad Moshiri.
Sean Dyche, who has been out of work since being dismissed by Burnley in April, is the other main contender to take over at Goodison Park. The Toffees hope to have someone in place by the weekend.
Bielsa is believed to have reservations about the composition of Everton's squad and its ability to play his trademark high-tempo style of football, as well as taking over in mid-season.
But the fact that he has flown to England to discuss the possibility further shows there is still interest on both sides. Dyche is also in the final round of discussions.
Should Everton get someone in place before the weekend they would have a week to work with the players before they welcome Premier League leaders Arsenal next Saturday.
Everton sacked Lampard after Saturday's 2-0 loss at West Ham left them 19th in the table with only three wins and 15 points from 20 games.
Lampard took over from Rafael Benitez last year in January after the Spaniard was sacked with the Toffees in 16th place, four points above the relegation zone.
They secured their top-flight status with a game to spare after a stunning fightback from two goals down to beat Crystal Palace 3-2 at Goodison Park.
In his first public post since being sacked, Lampard said he joined the club in a "tough moment" and "will always be proud of the incredible work and support from everyone involved" to stay in the Premier League last season.
"I will never forget the incredible night against Palace that we shared," he said on Instagram.
