In an interview with The Guardian, the 30-year-old has alleged that Sampson made the comment at the team hotel ahead of an international match against Germany in 2014.
Aluko told the coach that her family were travelling over from Nigeria to watch the game, to which the 34-year-old allegedly said that he hoped that her relatives would not "bring Ebola" with them.
The former Birmingham City Ladies star, who moved to England from her homeland of Nigeria as a young child, told the newspaper: "We were in the hotel. Everybody was excited. It was a big game. On the wall, there was a list of the family and friends who were coming to watch us and I just happened to be next to Mark.
"He asked me if I had anyone who would be there and I said I had family coming over from Nigeria. 'Oh,' he said. 'Nigeria? Make sure they don't bring Ebola with them.' I remember laughing but in a very nervous way. I went back to my room and I was really upset.
"It might have been easier to take if it was about me alone. Lots of things had been said about me over those two years but this was about my family. I called my mum and she was absolutely disgusted."
The Football Association was reportedly made aware of the comments in November 2016, but no action was taken against Sampson, who has been in charge of the England Women's team since 2013.
It is believed that Aluko was paid £80,000 to sign a confidentiality agreement, but has now been given the green light to talk about the remarks in the public domain.
In a separate incident, Sampson is alleged to have asked a mixed-raced England player how many times she had been arrested, but after an independent inquiry, which did not include an interview with the footballer, the coach was cleared.
Aluko represented her nation between 2004 and 2016.