Quinton de Kock was expected to lead but Cricket South Africa ruled him out of the job, saying they don't want to overburden the wicketkeeper-batsman.
The CSA's scheduled three-Test series with West Indies begins in July, are still trying to find out a solution for the problem.
"It's definitely not an easy journey being a Test captain but I think leadership is something that comes extremely naturally to me," Elgar said during an interview with Cricket South Africa (CSA) which was released on Monday.
"I've captained teams in the past, right up from school level to provincial level and at franchise level. I enjoyed it extremely and if I was asked to take on the captaincy, it's definitely something I would think long and hard about because it would mean a lot to me."
