During a two-legged Champions League qualifier between the Swiss side and Maccabi Tel Aviv last August, Salah refused to shake hands - instead changing his boots in one instance and bumping fists with his opponents in the other - while there were also comments attributed to him expressing an anti-Israeli sentiment.
Salah insisted the comments were fictitious and his actions were a result of external pressures.
Yet Mourinho has promised to nurture the 21-year-old once his move is complete to a club owned by Russian billionaire Roman Abramovich, who is Jewish.
"It's not a concern, because we know the club Chelsea is, I know the manager I am," Mourinho said.
"I know that Chelsea's capable of helping people and making people feel the way we think people should feel, which in football and in life to be open and respectful to every ethnicity, to every religion.
"People have the right to be what they are. Chelsea as a club and me as a manager we will be more than ready to help him."
The incident with Maccabi Tel Aviv took place under UEFA jurisdiction, but Press Association Sport understands anti-discrimination groups will be monitoring the situation.
Chelsea announced Salah's signing on Thursday, subject to the agreement of personal terms and the passing of a medical, with the forward replacing Manchester United-bound Juan Mata in the squad.
Chelsea plan to travel to Switzerland to meet Salah to speed up the finer details of the transfer as he is a non-European national and would need a visa to travel to London.
Mourinho said: "He's a non-EC player. He cannot come to England for his medical, so we have to travel to meet him and to have the medical, probably in Switzerland.
"Only after that can we do the paperwork to bring him to the country."
Mourinho lauded the qualities of Salah, who scored in each Champions League win for Basel over Chelsea earlier this season and at Stamford Bridge in the Europa League semi-final second leg last term. The performances also impressed Liverpool.
"I like the fact that he can play the same way our attacking players do, which is players with adaptability, to play right, left, or behind the striker," Mourinho said.
"He's young, he's fast, he's creative, he's enthusiastic.
"When we analysed him he looks the kind of humble personality on the pitch, ready to work for the team and to work and to adapt himself to a new life.
"We believe if he comes that he will have a similar process to the Willian one. Willian needed two or three months to grow up and to feel comfortable playing for us (following his August signing from Anzhi Makhachkala).
"We think with him and (Andre) Schurrle, (Eden) Hazard, Oscar and Willian we will be fine.
"We lose an experienced top player like Juan, but we bring in a young player with great potential.
"Willian did that transition from his previous world to the top world of football. Hopefully, if Salah comes, he can do the same.
"We are ready to help, we are ready to support.
"He has some experience in Europe, the Europa League, Champions League, so he's not a naive boy arriving in the jungle.
"Hopefully he comes and hopefully with the talented players we have around in the same positions we all can develop together."