Everton are playing at Anfield in the Premier League on Saturday, before travelling to Old Trafford on Tuesday in a crucial double-header.
A superb run of form that has seen Koeman's side lose just once in the top flight in 2017 has them sitting just six points behind fourth-placed rivals Liverpool.
And Koeman feels a win and a draw in the matches against Jurgen Klopp's men and United will give his team every chance of sealing a top-four finish.
"I believe that yes [four points would see us challenge for the top four]," he said to Sky Sports.
"We need to show it and depending on the next two results, Liverpool and then Manchester United away on Tuesday, that makes our possibilities until the end of the season more clear.
"If you look at Everton since the start of 2017, we have had a great run of results. We have had the best results of all the Premier League teams, but we know that Liverpool and Manchester United away are really tough games.
"But we had a draw away at Manchester City and in my time managing Southampton we won against all the big teams. This Everton team is not less quality than that Southampton team was and it is football.
"You need to believe and you need to believe in the quality of your team-mates and you need to show that aggression and control your emotions, because that is also a really important part of the Merseyside derby."
Key to Everton's hopes will be the performances of in-form striker Romelu Lukaku, the Premier League top scorer having netted nine times in his last six top-flight outings.
"Everybody knows he is one of the best finishers," added Koeman.
"But he is also improving in his other qualities, he is working harder than he did, he is running more than he did.
"We have the numbers and he is the number nine that every manager would like in their team."
Everton have not won at Anfield since 1999 and have not won any Merseyside derby since 2010.