Brendan Rodgers' men need the Hammers to beat league leaders Manchester City at the Etihad Stadium on the final day, which would mean a victory against Newcastle would give the Reds a first league title in 24 years.
And Lucas says he will be asking Carroll and Downing, who both moved to Upton Park from Anfield on permanent deals last summer, to do their former team a favour.
|14/1||Andy Carroll is 14/1 with Paddy Power to open the scoring against Man City|
"They have been here with us, working with us and I'm sure they will do their best to get points for West Ham first and if they can help us.
"But it's not in our hands anymore. Against Crystal Palace (in a 3-3 draw) for eight or nine minutes we just switched off and paid the price. We have to learn from that.
"We just have to improve and see if we can get any luck now until the end of the season, even if we know it is hard now."
Liverpool threw away a three-goal lead against Palace at Selhurt Park and Lucas admitted the team struggled to deal with such a calamitous defeat.
"It was very quiet in the dressing room after the game, probably the most quiet I have ever seen since I joined the club,” the Brazilian added.
"You could see on the faces of the players at the awards on Tuesday that we still hadn't recovered from the Palace game. The day after is always the worst."