43-year-old Cantlie was kidnapped in Syria in 2012 and has recently appeared in a series of videos as well as an article released by IS.
In a statement, the freelance journalist's sister Jessica said that the family had previously been in contact with his captors, but that the channel of communication had been lost and the new efforts to speak to the jihadists were ignored.
"We implore IS to reinitiate direct contact," she said.
"It is not true to say there has been no attempt to engage with IS. This is simply not accurate. This is frustrating for all parties, including those who are trying to assist us.
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"We had previously been in contact through a channel started by you. But this stopped for reasons best known to you," she said in the statement.
Cantlie, who is originally from Hampshire in England, has been seen in videos which have all followed the same format, with the freelance journalist sitting at a desk wearing an orange jumpsuit like prisoners at Guantanamo Bay against a black backdrop to address the camera.
Since August, IS has killed four Western hostages - US journalists James Foley and Steven Sotloff and British aid workers David Haines and Alan Henning.
Referring to the British victims, Jessica said: "Sadly like the families of David Haines and Alan Henning, before they were killed, our efforts at re-opening dialogue continue to be ignored by those holding John."
"We strongly challenge those holding John to return to your previously-opened channel, to which we continue to send messages and await your response so that in keeping with everyone's wishes, we can restart dialogue," she said.
He was rescued from kidnappers in 2012, but four months later chose to return to the country, where he was abducted a second time and allegedly sold on to IS.