Tottenham delivered on their pre-season hype by thrashing Southampton 4-1 on Saturday, while Chelsea also got off to a winning start as Jorginho's penalty earned a 1-0 win at Everton.
Recent acquisition Gabriel Jesus wasted little time in making his presence felt in his first start for Arsenal, opening the scoring in a 2-0 friendly win over Everton in Baltimore, Maryland, on Saturday.
Everton majority shareholder Farhad Moshiri has insisted the Premier League club is not for sale after talks with a consortium looking to take over at Goodison Park.
Tottenham are set to further bolster their attacking options with the signing of Richarlison from Everton in a deal that could rise to £60 million ($73 million), according to reports in the British media on Thursday.
Manchester City retained the Premier League title in the most dramatic of fashion as they hauled themselves back from 2-0 down to beat Aston Villa 3-2 with three goals in the final 15 minutes on Sunday.
Everton staged a dramatic fightback from two goals down to preserve their Premier League status with a 3-2 win against Crystal Palace, while Burnley climbed out of the relegation zone after a 1-1 draw at Aston Villa on Thursday.
Chelsea all but guaranteed a third-place finish in the Premier League after playing out to a 1-1 draw with Leicester City at Stamford Bridge on Thursday evening.
The Premier League title, a place in next season's Champions League and the battle for survival are all alive heading into the final week of the season.
Manchester City can put one hand on the Premier League trophy on Sunday should they beat West Ham to extend their three-point lead over Liverpool at the top of the table.
Kevin De Bruyne produced a four-goal masterclass as Manchester City thrashed Wolves 5-1 to move three points clear at the top of the Premier League, while Chelsea consolidated their top four place and pushed Leeds deeper into relegation trouble on Wednesday.