The British 27-year-old, who has a 30-3 record with 19 knockouts, is in line for a blockbuster with Mayweather (47-0, 26 KOs) in May.
Khan is calling on Mayweather - the WBA, WBC and The Ring welterweight title holder - to take him on, but the undefeated champion is also being pursued for a bout with Filipino superstar Manny Pacquiao.
Mayweather said Khan lacked the status to fight him, but the Briton - who earned a unanimous decision over Devon Alexander in Las Vegas on Saturday - slammed those claims.
"He's very silly to say that because I am one of the biggest names in boxing," Khan said.
"I'm bigger name than [Carlos] Maidana, I've beaten Maidana, we fought twice, so he's telling me I'm not a bigger name than Maidana.
"Victor Ortiz, who he [Mayweather] fought, and beat. I'm a bigger name than him.
"He's just avoiding me, he's looking for little excuses that he doesn't want to fight me."
Khan added: "After a performance like that [against Alexander], I think I've secured my fight with the best fighter in the world, which is Floyd Mayweather supposedly and I honestly believe he thinks I can beat him.
"I mean, he thinks I will do something - maybe that's why he's not comfortable fighting me but if he was, Floyd Mayweather's one of those guys who says 'I'll fight you and I'll fight you wherever'.
"I can't believe I've been calling him out for such a long time and he's not taking the fight. I don't know what I need to do to take the fight."
Khan, who said he wants a fight signed and sealed by January, claimed an alternative was to bypass Mayweather and tackle good friend and former sparring partner Pacquiao himself.
"I definitely think it will be a great fight with Manny Pacquiao because we've sparred together, we've trained together and we speak with each other on the phone quite a lot so it would be good to get that fight," Khan said.
"Look, it's a business, boxing and friends sometimes have to fight each other.
"I don't think it would have happened when we were training together but now we're not training together, I think it can definitely happen.
"We'll just have to put our friendship out the window for 45 minutes, see who the best fighter is."
Khan, of Pakistani descent, also donated the shorts he wore against Alexander, which had 24-carat gold threading, to the school in Peshawar that was attacked by the Taliban.
The shorts are estimated to be worth £30,000.