The Brighton defender was added to Gareth Southgate's 26-man squad after Trent Alexander-Arnold suffered an injury against Austria and was forced to withdraw from the group.
White was initially named on a 33-man long-list before being cut, when he began planning for a summer holiday to Portugal.
Now, having impressed against Romania and returned to the squad, the 23-year-old could well start in place of the injured Harry Maguire against Croatia.
"To make it to the final squad was unbelievable," said White.
"It's been very surreal. Obviously I didn't think this would be happening. I thought I'd be somewhere hot like Portugal [for the summer].
"It's mind-blowing. I didn't really expect any of it to happen. To make it to the final squad was unbelievable.
"I don't think it can be bad news when the gaffer is going to ring you. I was just waiting for the call. I don't think I've let it settle in yet.
"It's something I've never felt before. It's very difficult. It's news that doesn't normally come around. I was speechless and I felt amazing."