Matt Le Tissier has urged Southampton boss Ronald Koeman to sign striking reinforcements before the transfer window closes.
Downing, who left Liverpool to join West Ham earlier this month, made his first start for the Hammers in their goalless draw against Newcastle on Saturday.
The 29-year-old admits he was interested in a move to Newcastle, but says he felt that the Newcastle hierarchy blocked Alan Pardew pushing through a deal because of his age and his wages.
"There were things going on in the background and Newcastle enquired about me," Downing told the Daily Mirror.
"But it sounds like their owners are similar to Liverpool on transfer fees and wages and the player's age.
"That all comes into the equation, and I think that was obviously a problem for Alan Pardew.
"It was a move I was interested in and I looked at it, but when I spoke to Sam Allardyce he wanted me straight away.
"He pushed the boat out to get me, and there's nothing more you want as a player."