Not true.
Ilie grew up in a family of tennis players, thus he was very aware of it very early, even if he didn't play it except occasionally until 10 or so. His elder brother, Costel, was on the Romanian national team, while Ilie was a child.
In his book written with Ion Tiriac (and the great sports writer Ioan Chirila), Ilie writes:
"Until the age of 10, I played only soccer.
[then he started to get coaching from his brother who saw Ilie hitting occasionally and realized his amazing touch and from Costica Chivaru, a coach which was "paying" Ilie with sweets in order to make him play]
In 1959, at Cluj, in September, I won the national title for 13-14 years, with 6-4 6-2."
Ilie Nastase was born in 1946.
What is true however, is that about the age of 15-16, he had to take a decision between soccer and tennis, and fortunately for all of us, he stuck to tennis.