The Red Devils looked odds-on to claim an 11th Premier League title just two weeks ago following a blistering run of results.
However, they followed up their 4-1 defeat to Liverpool with a 2-0 loss at Fulham while Rafa Benitez's side crushed Aston Villa 5-0.
United's gap is down to just one point, although they still have a game in hand, but Alonso has conceded Sir Alex Ferguson's troops remain in pole position to re-claim their crown.
Difficult"It's going to be difficult because we can't afford to drop many points and we know that United have a very good team," he told Sky Sports News.
"They are the main favourites, of course, but we are going to be doing our best to be challenging until the final weeks of the season."
When questioned whether the pressure is on United, he responded: "Yes, I think so. They are the champions.
"They are in the best position and they have that advantage. We have to do our job and try to make things difficult for them."
No coincidenceAlonso believes is it no coincidence that the return to fitness and form of Steven Gerrard and Fernando Torres has been behind their recent run.
"At the moment the team is playing and defending well as a team," he added. "The partnership between Stevie and Fernando is working well for us.
"Of course [Torres] is really important for us. When he's fit he's a threat for all the defenders and that's a boost for us."
Liverpool have gone 19 years without the title, but Alonso is not letting his mind wander just yet despite having a real shout of ending their drought.
"No, absolutely not," he said, when asked if he was thinking of title glory. "Once it happens I'll be more than happy to think about it, but at the moment, nothing so far."