The powerful Nigerian international scored in his first game since returning to full-fitness against West Ham on December 22, and then started in the Boxing Day victory over Wigan at Goodison.
David Moyes would like to add fresh attacking impetus to his squad when the January transfer window opens, but with other areas he would also like to strengthen and a limited budget, Osman believes the 24-year-old could solve a problem for his manager.
He said, “Of course he could save the manager some money in January. That would be great for the manager I’m sure.
“He was a real handful for West Ham. He put himself about and was just as physical as they were. He surprised them with how strong he was and held the ball up well for us at times.
“If we can keep him fit he’s a great weapon to use whether it’s from the start of the game or coming off the bench.”
Anichebe has scored four times already this season, but has once again had his progress interrupted by injury – a streak of misfortune which Osman believes must change if the striker is to fulfil his undoubted potential.
“He’s shown when he’s fit he can be a handful for any side,” said the 31-year-old.
“But he has found it difficult to stay for long periods of his career so let’s hope he gets a bit of luck with that now.”
Osman was on the scoresheet himself against Wigan with a strike which deflected off Latics skipper Gary Caldwell on its way past Ali Al Habsi.
But after having already seen a legitimate header ruled out at West Ham in the previous match, the England international was in no mood to concede his latest effort.
“It’s mine,” he said. “I don’t know about ‘claiming it’; it is a shot that was going on target which has gone in the other side of the goal. As far as the rules go, it’s my goal.”
The goal helped Everton to six points so far over the festive period, and Osman is delighted with the timely shift in turning draws into wins.
He added, “It’s a great sign of the character we have. We were picking up draws and we needed to find a way of turning them into wins and we have done that in the last couple of games.
“Wigan made it difficult for us and played some really good stuff at times but the hectic Christmas period is all about picking up as many three points as you can and we did that.
“Chelsea have been strong over the years and increased their firepower this year. It is going to be difficult but we are full of confidence and we make it a tough game for anyone to come to Goodison, so hopefully we will do that again.”