Brendan Rodgers' men need West Ham to beat Premier League leaders Manchester City at Etihad Stadium on the final day.
That would mean a Liverpool victory at home to Newcastle would give them their first league title in 24 years.
And Lucas said he would ask Carroll and Downing, who joined West Ham on permanent deals in the off-season, to do their former team a favour.
Asked if he would send them texts, Lucas said: "Yeah. Of course.
"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 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."
Coughing up a three-goal lead in that 3-3 draw at Palace on Monday, and the 2-0 home loss against Chelsea the game prior, are results Liverpool will rue.
Lucas, 27, said his team's dressing room at Selhurst Park was the quietest he had encountered since joining the club in 2007.
"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 said.
"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."
