Everton are currently positioned in 12th place in the Premier League table.
HITC Sport understands that Middlesbrough chairman set to meet with the Spaniard on Tuesday.
According to reports, Redknapp could be about to take on the role himself.
The Spaniard took charge of the club for the first time for Monday night's trip to Leicester City.
West Ham United drew 1-1 with Manchester United at Old Trafford.
Hugo Lloris has helped Tottenham Hotspur keep a better defensive record than any other Premier League side this season.
Leicester City host Newcastle United this evening.
We continue our countdown of Aston Villa's greatest ever players with a tricky winger who set up the most famous goal in the club's history.
Dean Richards would spend his best footballing years elsewhere, but he developed into a Premier League centre-back during his time at Molineux.
The former Gunners captain claims supporters went too far with their protests against the long-serving manager.