The Rams will play in front of their own fans for the first time since March 2020 when they host league leaders Toulouse on Sunday afternoon.
That game is followed by a home clash with the league’s second-placed side Featherstone Rovers
Both clubs have yet to lose a game in the Championship in 2021, with Rovers also reaching the final of the 1895 Cup, and Greenwood admits things could not be much tougher for his side over the next two fixtures.
“We will need it [the support],” said Greenwood.
“Those two home games, against teams who are wiping the floor with everyone and who are undefeated, are obviously going to be very hard for us.
“We are going to need the fans right behind us. They are a knowledgeable and realistic bunch the Dewsbury fans, they just ask for the lads to put everything in.
“I am sure the lads will do that and I am sure the fans will get right behind us.”
Dewsbury have started the season with a difficult run of matches and by the end of the month they will have played the entire top half of the table, barring Batley Bulldogs.
“We had a pretty tough start to the season,” Greenwood added.
“Whoever put this run of fixtures together is obviously not a Dewsbury fan because we had to go to London and then play Toulouse and Featherstone.
“It is a little tough run and we are doing it tough as a squad as well due to people leaving and people getting injured.
“It is a bit of a firefighting job at the moment but there are probably a lot of clubs in similar situations as well.”
Meanwhile, Dewsbury player James Thornton has joined North Wales Crusaders on a season-long loan.
He made his debut for the League 1 club in their 36-12 victory over Coventry Bears last weekend.
He said: “I’m quite familiar with League 1 and every time I’ve played Crusaders, it’s been a tough game.”