Warrington was knocked out in the ninth round in a massive upset at the SSE Arena, Wembley.
The 30-year-old was overpowered by the heavy-handed Lara, who had barely fought at elite level before his victory.
The defeat leaves Warrington's hopes of fighting for a world championship this summer in ruins.
Joshua is another fighter who knows what it's like to be knocked out by a Mexican.
He lost his heavyweight belts to Andy Ruiz Jr in June 2019 in a similarly surprising result.
The 31-year-old followed up on that defeat with a composed victory over Ruiz six months later.
Joshua has sent his support to Warrington via FaceTime.
Revealing the call on Twitter, Joshua said: "You know what you gotta do. I got your back champ."
Warrington also posted about the call, saying: "Boxing school with Anthony Joshua. Lesson 1 - bouncing back after getting KO'd by a Mexican. Thanks for the call big man."
The Leeds-born fighter was scheduled to face WBA champion Xu Can in April, but that fight will now likely be off the cards.
Warrington's next fight could be a rematch against Lara, as he aims to avenge the first defeat of his career.
He reigned as IBF featherweight world champion for two years after beating Lee Selby by split-decision in 2018.