The fifth-round replay is scheduled for Upton Park on Tuesday, March 7, but Bolton could be in UEFA Cup action two days later.
That depends on them defeating Marseille in the return leg in France next Thursday with the scores locked at 0-0, and a place in the last 16 up for grabs.
The winners are likely to meet the Russians of Zenit St Petersburg, who hold a 2-0 lead against Rosenborg.
Should Bolton progress they will request the FA Cup replay to be switched to March 6 as they also have league games against Newcastle on March 4 and West Ham on March 11 to contend with.
Allardyce said: "I would rather be in the UEFA Cup than not be.
"Even though it would be difficult being in two competitions in one week that's what I would like to happen.
"Our disappointment is that in the Marseille game and this one we haven't done the business at home.
"We are just lacking a cutting edge at the moment and we are paying the price.
"Everything else was great today to win the game but we just could not score."
Gary Speed and Kevin Nolan both had good chances to make the breakthrough but Bolton had to settle for a share of the spoils for a fifth successive match.
Allardyce was at a loss to understand why they were not awarded a goal in the 64th minute when Stelios Giannakopoulos appeared to be onside.
"We've had to take that one on the chin," said the Bolton boss. "But when you see it is the wrong decision it makes you very disappointed indeed.
"When you look at when the ball was actually played he is level at the very most.
"We have just got to keep playing this way and the draws will turn into wins again."