Sampson was dismissed by the Football Association in 2017 following the emergence of “inappropriate and unacceptable behaviour” in his previous role as manager at Bristol Academy.
But the 36-year-old had also been accused of bullying and racism by female England players Aluko and Spence.
Aluko claimed Sampson told her to make sure her Nigerian relatives did not bring the Ebola virus to a friendly against Germany at Wembley in November 2014.
Sampson denied that claim, along with another allegation that he asked mixed-race Chelsea midfielder Spence if she had been arrested before.
The FA said the allegations were investigated, and Sampson was cleared of any wrongdoing.
But Sampson, who reached a financial settlement with the FA over his dismissal earlier this month, now says he should have handled the claims differently.
Speaking to The Times, he said: “I answered based on my recollections and what I felt was true to my character and the way I use language.
“I believed strongly that the way the team worked, if I had said something that offended anyone it would have been brought up at the time.
“I was wrong. Ultimately, two players were offended by language I used. For that I apologise to Drew and Eni unreservedly.
“I certainly should have played a much more active role to ensure they (the complaints) were handled more respectfully. That’s a big regret I will always have.”
Sampson was sacked one day after England beat Russia 6-0 following a safeguarding investigation into his time at Bristol, where he had a six-month relationship with a player aged over 18.