Elgar has played 63 Tests for South Africa and scored 3,888 runs at an average of 38.12.
Elgar, the team's longstanding opener in the red-ball version, confirmed his desire via an interview with Cricket South Africa (CSA), released on Monday (May 25).
The 32-year-old joined young Aiden Markram in the list of interested applicants for the job.
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"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.
"I've captained teams in the past, right up from school level to provincial level and at the 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."
Quinton De Kock has started off well in white-ball cricket after taking over from Du Plessis. He would certainly have been a candidate for Test skipper's slot as well, but for his already excessive workload, given the 27-year-old is also South Africa's all-format wicketkeeper-batsman.
Proteas are due to next play Test cricket on their tour of the Caribbean in July, which is highly uncertain at this stage because of the COVID-19 pandemic.
Elgar has twice stood as captain in absence of Du Plessis, once in a defeat on the 2017 tour of England and at home in a victory against Pakistan in early 2019.
(Inputs from Reuters)