Rodgers' men saw their FA Cup hopes ended in a 2-1 semi-final defeat at Wembley last weekend, with Aston Villa turning the game around after Philippe Coutinho had put Liverpool ahead.
Liverpool are seven points adrift of fourth-placed Manchester City and the last UEFA Champions League qualifying berth in the Premier League, with Klopp's decision to leave Borussia Dortmund at the end of the season thought to have aroused the interest of a number of clubs.
"When you are at a club like Liverpool, there will be lots of names mentioned," said Rodgers. "It goes with being at a big club.
"I don't think there's anyone better [than me to manage Liverpool]. I have a strong relationship with the owners.
"This is a club with ambition, that's why we have to strive to be the best. We want to be winners.
"I'm motivated to make sure we progress and make the next step."
Rodgers takes his side to West Brom on Saturday, with Liverpool having lost three of their last five matches.
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