The Greenock club had to forfeit the away match against Stranraer on Saturday 3-0 due to "multiple players returning a positive Covid-19 test", with their second Group G fixture at home put in doubt with ticket sales suspended on Monday.
A statement on Morton's website read: "The club can confirm that discussions are still ongoing with the relevant bodies regarding tonight's game against East Kilbride
"However, in the interests of supporter safety, we have taken the decision that, should the game go ahead, it will take place behind closed doors.
"Cappielow is currently being deep cleaned and, due to this, and the absence of key members of staff due to Covid issues, we do not feel it would be responsible to allow supporters into the stadium tonight.
"This is obviously extremely disappointing, but the safety of supporters, staff and players is our priority.
"We will be providing a stream for tonight's game as planned at £12, however the stream will not have commentary attached due to the circumstances.
"We will be refunding all tickets purchased for tonight's game, but this may take us a bit of time.
"Due to the above staffing issues, we apologise this cannot be done straight away but will provide a further update in the next couple of days on how to claim your refund.
"Once we have definite confirmation as to whether the game will go ahead, we will provide a further update to supporters."