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Manchester City boss Pep Guardiola says his side will have to "fight until the end" in the race to secure a Champions League spot after slipping up on Saturday.
City could have taken a huge step towards sealing a top-five spot, but drew 0-0 at rock bottom Southampton.
They are now only two points above Newcastle and Chelsea - who meet on Sunday - plus Aston Villa, who beat Bournemouth on Saturday evening.
Seventh-placed Nottingham Forest, who play relegated Leicester on Sunday, are four points behind City.
After the Saints draw, Guardiola said his side had three cup finals left - including next weekend's actual FA Cup final against Crystal Palace.
"I didn't expect differently from a month ago, that it's a fight until the end," he said.
City could find themselves outside the Champions League places before they play in the league again (against Bournemouth on Tuesday, 20 May) - because most of their rivals will have played once or twice before then.
Only champions Liverpool have sealed a Champions League place so far but second-placed Arsenal, who visit Anfield on Sunday, will hope to soon join them.
There will be nine Premier league teams competing in Europe next season - up from the usual seven - but things could change in the final weeks of the campaign.
Crystal Palace, who are 12th, would qualify for the Europa League if they win the FA Cup final.
Brentford, Brighton, Bournemouth, and to a lesser extent Fulham, are all vying for eighth spot which might be enough.
The Bees are in eighth, above Albion on goal difference, two points ahead of the Cherries and four clear of the Cottagers.
Brentford boss Thomas Frank, after a 1-0 win at Ipswich, told BBC Sport: "We have got the momentum which is important.
"We are close to finishing eighth which would be the best position ever in the league. We are laser focused on what we can do. We have given very little away but still created a lot.
"We just need to push and enjoy it, the next two games, and then reset our target."
Manchester United and Tottenham meet in the Europa League final with the winners qualifying for the Champions League.
Statisticians Opta give Arsenal a 99.7% chance of qualifying for the Champions League.
They give a 91.2% chance for City, 69.4% hope for Newcastle and 67.9% for Chelsea.
For Forest it's 37.8% and for Villa it's 34.2%.
Liverpool: Arsenal (H); Brighton (A); Crystal Palace (H)
Arsenal: Liverpool (A); Newcastle (H); Southampton (A)
Manchester City: Bournemouth (H); Fulham (A)
Newcastle: Chelsea (H); Arsenal (A); Everton (H)
Chelsea: Newcastle (A); Manchester United (H); Nottingham Forest (A)
Aston Villa: Tottenham (H); Manchester United (A)
Nottingham Forest: Leicester (H); West Ham (A); Chelsea (H)
Brentford: Fulham (H); Wolves (A)
Brighton: Liverpool (H); Tottenham (A)
Bournemouth: Manchester City (A); Leicester (H)
Fulham: Brentford (A); Manchester City (H)
There will be nine English teams in European competitions next season - with six in the Champions League.
England got an extra spot because of their teams' performances in Europe this season - and another one because of the make-up of the Europa League final.
Manchester United and Tottenham meet in the final on Wednesday, 28 May in Bilbao - with the winner guaranteed a Champions League spot.
But since neither side can qualify for Europe through their league position, that will be a bonus spot.
A maximum of three Premier League teams could be competing in the Europa League next season.
The team who finish sixth and the FA Cup winners are the two currently due to get a spot.
However, if the cup winners have already secured a European place, it will then go to the next highest placed side who have not qualified for Europe.
A third place could be awarded if Chelsea win the Conference League and fail to qualify for the Champions League.
The Blues face Real Betis in the final on Wednesday, 21 May in Wroclaw.
If Chelsea beat Betis and finish in the top five, there would be two English teams in the Europa League and one in the Conference League.
There would usually be one Conference League place awarded to a Premier League team.
As things stand that will go to Carabao Cup winners Newcastle, but that could change if Eddie Howe's side qualify for the Champions League or the Europa League (so finish in the top six at least).
In that scenario seventh spot would be a Conference League place if Crystal Palace win the FA Cup.
If Manchester City win the FA Cup and qualify for Europe through the league, then the team in eighth place will head into the Conference League.
However as mentioned above, if Chelsea win the Conference League but do not finish in the top five then there will no English teams in next season's competition.