The 29-year-old striker is the sort of physical presence up front that Allardyce loves in a forward, and asked on ESPN whether it'll be a hit or a miss for him, Robson suggested it will happen.
He said: "It would suit Sam Allardyce because he's a big player and hasn't got a great deal of pace. He's good in the air and will back into centre-halves, making it difficult.
"Long term I don't think it's a good move, but short term, with Sam Allardyce's style of play, I'll go with a hit."
Deeney, who has scored twice in 10 league appearances this season, wouldn't offer a huge deal in regards to goals and that is precisely what Everton need in Janaury - a goalscorer.
Stoke, in 13th, have only scored one less goal than the Toffees' haul of 19 in 15 matches which, for a side that spent £150 million in the summer, isn't a brilliant return by any means.
Deeney is valued at around £10 million according to Transfermarkt and it's probably a bit too much to pay for a player who is more a professional nuisance than a goalscorer.
Deeney has 25 goals in 85 top-flight games overall.