The 26-year-old moved to Goodison Park from Besiktas in the first week of the mid-season window in a deal worth £27m (Sky Sports).
And that interest has since been substantiated by loan technical coach Eddie Newton, who was sent by Chelsea to watch the hitman prior to his move to Merseyside last month.
"He is a very good player, I watched him at Besiktas and was impressed," he said, according to the Liverpool Echo. "In football you rarely sign a player and expect immediate results, especially in January.
"He will need time. The Premier League is the fastest in the world and it takes time to adjust. Cenk is settling in and adapting. I really think he is going to do well in England but needs to be given a chance and time to settle in first."
Tosun has endured a tough start to life in England, starting just two of the Toffees' five league games since his arrival and was an unused substitute in the win over Crystal Palace at the start of the month.
Manager Sam Alladyce has attributed the cold weather since Tosun's arrival as the reason behind his slow start but has backed him to return to his best form, having netted 14 times for Besiktas during the first half of the campaign.
"He is struggling with the pace of the Premier League which happens to more players than it doesn't who come in in January," Allardyce said, according to The Daily Telegraph.
"He was up to scratch when he was playing but there is more demand at this level to get to the level and show the ability he has got. There is no doubting his goalscoring ability but you have to have the capabilities to get in those positions to score those goals and that is the hard bit.
"When you are on the grass in short sleeves and shorts and a pair of boots players don't want to stop training and don't want to stop playing football, in fact we have to stop them. That is what you get when you get out in the right climate."
Meanwhile, though Tosun moved to Everton over Chelsea, the West Londoners were able to sign a new striker during the January window with France international Olivier Giroud joining from Arsenal on deadline day.
The move triggered the departure on loan of Michy Batshuayi, who has joined Borussia Dortmund on loan for the remainder of the 2017-18 campaign.