Ole Gunnar Solskjaer's team were reportedly interested in signing the Denmark international before Thursday's transfer deadline, but United ended talks due to the player's preference to move to La Liga.
Speaking to BBC Radio 5 Live (h/t Liam Prenderville of the Mirror), Scholes said he was delighted when Eriksen was linked with a move to Old Trafford, but he believes his former team have sufficient quality to break into the Premier League top-four:
"I was quite excited when Christian Eriksen got mentioned.
"He could have made a big difference to United in the way [Kevin] De Bruyne plays for [Manchester] City, creating chances and scoring goals.
"But they haven't got that, and they have some good players, so the target is top four, and I think they will make it."
United ended summer preparations with three new signings. Daniel James was bought from Swansea City to bolster the attack, and Aaron Wan-Bissaka was recruited from Crystal Palace to solve United's right-back issue.
Scholes added he thinks goalscoring could be an issue for United after the departure of Romelu Lukaku to Inter Milan. The Belgian was the club's recognised specialist as a No. 9, and Solskjaer now needs to convert one of his versatile forwards into the role.
"It is a worry for me, the centre-forward position," said Scholes. "They have got players like [Marcus] Rashford and [Anthony] Martial, [Mason] Greenwood, but are they really No. 9s? We have to wait and see. I think they are flipping between being a wide player and being a centre forward. They are yet to prove they can score 25-30 goals a season."
United also have the option of playing Alexis Sanchez through the middle, with the Chilean often used as a striker during his career at Arsenal.
Solskjaer has attackers he can mould into his system, with pace to burn up top, but a lack of creativity could be the club's primary issue during the new campaign.
Paul Pogba has the ability to feed the forward line, but the France World Cup-winner has a lack of support in terms of quality in and around the midfield positions. Angel Gomes could be United's wild card this season, providing Solskjaer with a burgeoning talent who can create in a similar fashion to Eriksen.