The 28-year-old's future with the Reds has been questioned after a lacklustre start to last season, and a possible switch to Upton Park has been mooted.
West Ham assistant manager Neil McDonald welcomed the prospect of Downing arriving at the club, but the former Middlesbrough and Aston Villa man says he is determined to fight for his place at Liverpool.
|16/1||Liverpool are 16/1 with Coral to win the Premier League
"I want it to continue. It's why I came here - to play in a successful team. I want that to continue as long as possible."
Liverpool are currently in Indonesia for the first leg of their pre-season summer tour and the players have been mobbed by fans every step of the way.
One of the 1000 fans who greeted the squad as they landed on Wednesday held aloft a banner with the words 'Stay here SD19 - don't go away', and Downing feels this support makes the difference to players.
"It's brilliant to have people behind you, the fans," he added. "When they're behind you, you want to do well for them and play well.
"We had a bit of a laugh about it on the bus but all in good spirits.
"There were great scenes and the lads are happy to be here.
"They were following the bus and chasing after it on motorbikes. I've not seen that at any of the clubs I've been at before. You can't doubt the support of this club worldwide."