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.
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 see what negotiations can be done."
Carroll's time in London has been blighted by injury, but he is now fully fit and judging by this performance, it will only be a matter of time before he earns an international recall after Roy Hodgson omitted him from his squad for the double-header against San Marino and Montenegro.
"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."
West Brom were poor throughout the 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 about his red card for petulantly kicking the ball at Gary 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 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. He probably did it out of frustration."
- Sports & Recreation