The midfielder was a firm favourite during his six and a half years at the club before leaving to join Arsenal last summer, and the Gunners travel to Merseyside in the Barclays Premier League on Wednesday evening for a clash between two of the country's in-form teams.
Round said: "He was a big favourite here and I think the Everton fans are very knowledgeable and respectful towards the senior players who have played for the club. I think they'll give him a good reception.
"He was a tremendous player for us. His attitude when he played for Everton was never in question, his ability was second to none and I think the Everton fans respect that.
"We do welcome him back and obviously we'll be fighting hard to make sure he doesn't have an influence on the game."
Arteta has settled well into Arsene Wenger's side, playing 23 times in the Premier League this season and netting six goals for the Gunners.
"They were big boots to fill when he went there as a sort of replacement for (Cesc) Fabregas but he's got the ability and the temperament to be able to handle that stage," said Round.
"It was at that stage of his career where that was the next step for him and when he left we wished him well."
Both teams will take confidence from their recent form, with Arsenal having won five league games in a row while Everton's 3-0 derby defeat by Liverpool last week was their first loss in any competition for two months.
In that time they have beaten Manchester City, Chelsea and Tottenham at Goodison Park, and Round said: "We've had a really good run of form against the big teams of late so we are looking forward to it.
"It should be a cracking game because I think Arsenal are currently top of the league on current form, they've won five on the bounce in the Premier League, we've lost one in 11 so it should be a really good game, and one both teams will go out and try hard to win."
Everton have a number of injury worries, with the game expected to come too soon for Darron Gibson, Jack Rodwell and Seamus Coleman, but Steven Pienaar is available again after being cup-tied for the FA Cup draw with Sunderland on Saturday.
Meanwhile, Round is confident Everton's medical checks are thorough enough following Premier League chief executive Richard Scudamore's comments on Tuesday that procedures would be reviewed.
The announcement came four days after Fabrice Muamba collapsed with a cardiac arrest during Bolton's FA Cup quarter-final against Tottenham.
Everton will hold a minute's applause prior to Wednesday's game to show support for Muamba, who remains in intensive care at the London Chest Hospital.
Round said: "I'm no medical expert but the guys assure me we screen the players every pre-season without fail. We adhere to the strict guidelines given by the Premier League and the FA.
"I'm sure now the whole system will be in review just to try to improve it a little bit more.
"From the club's perspective we obviously wish Fabrice the speediest recovery. The signs are looking good we believe from what we've heard and that's fantastic."
Everton will offer extra screening to players should they request it, although none have so far, and Round praised the way the squad have coped with the situation.
He said: "There was a little bit of shock and disbelief that this could happen to a fit young person and obviously people were saying 'if it could happen to him, it could happen to anybody'.
"But by the next day there was just an overwhelming show of support towards him. There were a lot of people asking how they could get in touch and what could they do, and I think you'll have seen that right the way through football."