The Spain midfielder admits United are still favourites to retain their Premier League crown but the events of the past few weeks have certainly opened up the race.
Liverpool's sensational 4-1 victory at Old Trafford was followed by United losing at Fulham the following weekend, while Rafa Benitez's men thumped Aston Villa 5-0 to close the gap at the top to just a solitary point.
Earlier in the campaign Liverpool also stole bragging rights at Anfield as they claimed a 2-1 victory on Merseyside.
United still have a game in hand on their North West rivals, but Alonso sees no reason why a 19-year wait for the title could not be sated this season.
"We can win the Premier League. It is far more likely than a month ago when we were a lot of points behind," he told the Daily Star.
"Liverpool have demonstrated this season that they are superior to United in the two matches between the teams.
"Nobody could dispute that, but now it is a real race for the title.
"It is not impossible to imagine that we could win all of our next eight matches while United lose two.
"But we know that we need to remain calm and to play without any real pressure or fear.
"United know what it is like to play under pressure, but they have shown there are some cracks in the last two games, and we will be ready to pounce.
"Every game is going to be vital from now on."
Alonso concedes Liverpool must win their remaining eight league games in order to have a realistic chance of the title.
"In our dressing room we are trying to remain calm. We know we need to take 24 points from a possible 24 so we need to be ice-cool about it. That is our challenge," he added.
"Mentally we are prepared to fight for the league and Champions League.
"United are defending champions and I feel they will be more nervous.
"It could be a very emotional end to the season for us and after such a long time to win the league would be fantastic.
"I feel we deserve it by the way we have already played this season, particularly in the big games."