The 20-year-old joined the Alex on loan in the summer and has made a big impact in League Two.
He has scored seven goals and provided four assists but his temporary deal expires in January.
“I’ve spoken to Chelsea and they say it is down to us”
Crewe boss Steve Davis
Crewe boss Steve Davis has held talks with the Premier League leaders and believes Kiwonya’s deal should be extended.
“I’ve spoken to Chelsea and they say it is down to us,” Davis told Crewe’s official website.
“We have to go back to them with something and perhaps do a little more in terms of the deal financially.
“At the moment we are paying a huge amount.
“We are paying what we can afford and maybe we will have to do a little bit more on that.
“The ball is in our court though and if we can do it then there is no reason why Alex can’t stay with us.”