Scott strengthened his position in Thomas Tuchel's England plans with a standout display in the 7-0 win over Croatia on Saturday.
The 23-year-old played 70 minutes alongside Manchester City's Elliot Anderson and earned a glowing assessment from Tuchel, who believes the emergence of another central midfielder gives England greater variety.
Chelsea remain particularly interested after having more than one bid, understood to be worth in excess of £60 million, rejected by Bournemouth during the summer, reports the Telegraph.
The London club are expected to continue looking for midfield reinforcements after Enzo Fernandez's move to Manchester City, with another approach for Scott possible in January or next summer.
Bournemouth have insisted they do not want to sell in January, but Scott will have only one year remaining on his contract at the end of the season.
The club have attempted to negotiate a new deal with the midfielder, although there has been no progress and both parties have agreed to concentrate on the current campaign.
Chelsea are not alone in monitoring the situation. Liverpool have shown interest under Scott's former Bournemouth manager Andoni Iraola, while Arsenal are also keeping an eye on him.
Tottenham Hotspur, Manchester United and Manchester City all invested heavily in midfield during the summer, potentially reducing the number of immediate alternatives for Scott.
His latest England performance has nevertheless strengthened his case for a bigger role at international level.
“Of course and he does, we need performances like this. This is just the nature of it and we will not survive on two central midfielders,” Tuchel said when asked whether Scott was competing with Declan Rice for a place.
“The more the better and it’s always about the guys who are in camp. Declan knows about his importance and he’s one of the very best midfielders in Europe. Nothing will change that.”
Scott also impressed alongside Anderson, with the pair having previously played together during England Under-21s' European Championship triumph.
“We work really well together. We saw that last year at the under-21 Euros and we’ve been speaking about it ever since, playing in this midfield together,” Scott said.
“I love playing with Elliot, he makes life so easy for me as his partner in there. He’s probably one of the best midfielders in the world at the moment.”
Scott had missed out on Tuchel's World Cup squad despite being included in the pre-tournament training camp.
The experience has not diminished his motivation.
“It gave me more hunger. But I’ve said there’s so many top players for this country, it’s a tough squad to get into.”
Chelsea therefore face competition if they decide to return for Scott, but their summer pursuit means they already have a significant head start. Bournemouth's contract situation could make the midfielder an increasingly attractive target as the season progresses.