The Everton youngsters made their debuts this season, and the Blues are apparently keen to tie them down.
It was revealed this week that Everton are keen to offer Davies, who made an impressive Premier League debut in the final day win over Norwich City, a new long-term deal at the club.
The Toffees are not stopping there though, and will continue to invest in youth players moving forward.
That will see them move to secure some of their brighter talents on improved deals, and both Dowell and Connolly are among those who they will make offers to, reports the Liverpool Echo.
Both players also made their debuts this season, with Dowell another to star in that final game win over the Canaries.
Dowell only signed his first professional deal with the club last March, but that expires at the end of next summer.
Connolly’s deal also expires at that time, so they will likely be the first to be offered new deals to ensure they aren’t snapped up by anybody else.
The deals are hardly surprising from the Blues given their desire to develop their own talent over the years.
And it appears that will be continuing, despite the recent influx of money from Farhad Moshiri.