Koeman also praised striker Lukaku, who equalled the club record of 60 Premier League goals set by Duncan Ferguson, who is on his coaching staff. "We know one of his big qualities is scoring goals", he said.
Despite the ongoing speculation, Dutchman Koeman - who is less than 12 months into a three-year deal following his arrival from Premier League rivals Southampton - is not interested in renewing his contract at Goodison Park.
Four points separate Everton from Manchester United in the Europa League hunt while they are six points adrift of Arsenal in the final Champions League spot.
Ronald Koeman's first season has been better than expected and the new signings have gelled admirably along with the existing crop of players.
Stats show that both sides have met 144 times in the past, with the home team slightly better than their guests with 67 wins and 23 draws.
But Everton - who Moyes led for 11 years before leaving in 2013 - did enough to get the points that keeps Sunderland in serious trouble courtesy of Idrissa Gueye's opener five minutes before half-time and Romelu Lukaku's record equalling second ten minutes before the end. Permission to use quotations from this article is granted subject to appropriate credit being given to www.theeagleonline.com.ng as the source.
Amal Clooney shows off growing baby bump in STUNNING dress
The couple made their first appearance on a red carpet since the announcement of the pregnancy of the lawyer on February 8. Dressed in a white strapless gown with a feathered skirt in fading shades of gray, Amal was the definition of radiance.
"I just believe that we'll get a good run somewhere along the line".
"Earlier in the season, I was a little bit inconsistent and I think I've got to a point now where I understand what I need to do, in games and preparing for games".
Moyes whisked his first team squad off for a short break in NY in the wake of their 4-0 thumping by Southampton at the Stadium of Light a fortnight ago which left them firmly entrenched in the bottom three.
"I think today's game was always a tough game for us".
"Football's like that as you have to test yourself if you want to improve". We have to find a place to score and move the ball well. "We need to stop him".
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