Fifth-placed Liverpool's lead over Everton narrowed to five points over the weekend after the Reds crashed to a 3-1 defeat against champions Leicester City on 27 February.
The loss was the Reds' fifth in their last seven matches in all competitions and they have picked up just six points in the league in the 2017 calendar year.
In contrast, Everton are on a nine-game unbeaten run in the league and beat Sunderland 2-0 at Goodison Park on 25 February, with Idrissa Gueye and Romelu Lukaku both finding the net.
Carragher believes the Toffees have made strides under Koeman's leadership and said that a sixth-place finish would be a good achievement in the Dutchman's first season at the club.
"For Everton, there's a feel-good factor about the place," the former Liverpool defender told Sky Sports.
"As a Liverpool fan you have to be worried about Everton coming at this moment. There's a derby game at Anfield in a few weeks, Everton haven't won there in a long time but if they were to get the three points it would be really tight towards the end of the season.
"Everton have been in seventh for a while now but there was always a big points gap between them and sixth. They've bridged that gap now, they could actually look at it and think: 'Can we get sixth position?'
"If Ronald Koeman can get them to finish ahead of one of the so-called big six, I think it would be a really promising start to his Everton managerial career."
Carragher said the emergence of Tom Davies and the January signing of Morgan Schneiderlin from Manchester United had revitalised the Everton midfield.
"They made good signings, and the confidence going forward with the manager bodes well for the future," he added.