Murray has a bye in the first round of the tournament so will not be in action until Wednesday, by which time he is hopeful of having returned to full fitness.
"When I had the injury, I had to take two-and-a-half weeks off serving. When I started serving again, I had to progress very slowly, but in the last couple of days I've been serving pretty much close to the speed that I would normally," he told The Independent.
"My elbow has reacted well, so I feel good about it. I will have had pretty much five days before my match of serving at the right speed, so I think it will be fine.
"I don't feel more tired than usual. I'm training really well. I've got lots of tournaments coming up, so I'm looking forward to it. Obviously I have some work to do to push myself back up in the rankings again this year. That starts this week."