Carroll has endured a difficult time in east London since he arrived on loan from Liverpool last summer, but he looked back to his menacing best on Saturday, scoring a fine brace which eased any lingering fears of relegation for Allardyce's men.
The 24-year-old fired a bullet header past Ben Foster to put West Ham 1-0 up and he then sealed the victory with one of the goals of the season following Gary O'Neil's strike.
Carroll watched the ball come over his shoulder after James Collins had sent a long punt up field and he volleyed past Foster with his right foot from a tight angle.
When asked whether he would like to retain the 6ft 3in striker's services next season, the West Ham manager said: "The answer to that would be yes.
"There is always a possibility. We will look at the end of the season and we will se what negotiations can be done."
Carroll's time in London has been blighted by injury, but he is now fully fit and judging by Saturday's performance, it will only be a matter of time before he earns an international recall.
Roy Hodgson omitted Carroll from his squad for the double-header against San Marino and Montenegro, but when the England manager comes to watch the footage of Saturday's game at Upton Park, he will be impressed, Allardyce reckons.
"That was the best we have seen of him here," Allardyce added.
"Because he has put two goals in, on top of a good performance.
"I hope this kicks him on to being a regular scorer for us in the next eight games.
"He is in a good frame of mind now and finally after a long wait his fitness is good and that's why you are seeing these performances and the goals are coming."
West Ham are now two points short of the 38-point mark Allardyce thinks the club will need to avoid relegation.
The only disappointment for the West Ham boss Saturday was the fact that he is likely to be without George McCartney for the rest of the season after he hobbled off with the same knee injury that had kept him out of the previous four months.
"I would have thought (he will miss the rest of the season)," Allardyce said of the left-back.
"It's a recurrence of the same injury, his medial ligaments, which is a very big disappointment for him and for us.
"That was the only downside for us today [Saturday]. Otherwise, it was a great day for us.
"It was a great performance and it takes us ever closer to that safety margin."
West Brom were poor throughout Saturday's encounter. Aside from a deflected Romelu Lukaku free-kick in the first half, Jussi Jaaskelainen barely had a save to make.
The only time the Baggies beat the Finn was three minutes from the end courtesy of a Graham Dorrans penalty.
West Brom boss Steve Clarke said: "I can have no complaints about the result. West Ham deserved to win.
"It was a difficult afternoon but we haven't had too many of those this season."
Clarke will speak to Youssouf Mulumbu on Sunday about his red card for petulantly kicking the ball at O'Neil in injury time.
"I said to the group after the game that they must control their frustrations," the Scot said.
"It doesn't affect today's [Saturday's] game but it affects our group and our squad and the next three games.
"We have a small group and it's a big loss for us.
"The time to speak about it is tomorrow [Sunday]. He probably did it out of frustration."