Tottenham boss Mourinho broke off from answering football questions in a pre-match news conference, saying the loss of the husband of Queen Elizabeth II would be felt far beyond British shores.
The Portuguese head coach said: "I’m sorry, I will answer your question. I just read some sad news about Prince Philip. I would like to express my condolences to the Royal Family and to be very honest, and say I have deep, deep, deep, the utmost respect for the Royal Family.
"I believe that it is not just this country that is going to be sharing these feelings, because I'm not English and I know that many like myself will have the utmost respect."
Mourinho, who has spent much of his coaching career in England, added: "I feel sad for the departure of Prince Philip on a personal basis, because I have only positive feelings for the meaning of the family. I feel very sorry. But unfortunately his life ends and we have to keep going."
Former England striker Lineker, now a renowned broadcaster, wrote on Twitter: "Sorry to hear that Prince Philip has passed away. Served this country over many, many decades. Sincerest condolences to Her Majesty, The Queen and The Royal Family. RIP Prince Philip."
The Premier League said it was "deeply saddened" by the news, adding: "As a mark of respect, players will wear black armbands and there will be a minute's silence before kick-off at all Premier League matches played tonight and across the weekend."
The Football Association, which the duke served as president from 1955 to 1957, said it felt "Immense sadness".
The FA asked royal and government authorities whether football should be allowed to continue this weekend and was given the green light.
"Following the sad passing of His Royal Highness The Prince Philip, Duke of Edinburgh, the FA has sought guidance from the Royal Household and Her Majesty's Government," the FA said. "We can confirm all football fixtures this evening and over the weekend (10-11 April) can continue at the discretion of competition organisers.
"The FA will be recommending that black armbands are worn and a minute's silence is observed before matches are played, including tonight's England women's international fixture in France. As a mark of respect, all flags at Wembley Stadium and St George's Park will fly at half-mast and the Wembley arch will also be lit."
Organisers of Saturday's Grand National said the race would go ahead at Aintree, preceded by a two-minute silence on the course, with "jockeys invited to wear black armbands".
Wimbledon, London's tennis grand slam, said it wished to "convey our deepest sympathy to Her Majesty The Queen and The Royal Family".
Imran Khan, the prime minister of Pakistan and former captain of his country's cricket team, wrote on Twitter: "My condolences on the demise of HRH Prince Philip, Duke of Edinburgh. Britain has lost a wise elder who was imbued with a unique spirit of public service. His role in promoting Pakistan-UK relations will always be remembered."
Prince Philip was a two-time former president of the MCC and a noted cricket enthusiast, playing what was described as an "instrumental" role in introducing trophies for winners of the County Championship.
Ian Watmore, chair of the England and Wales Cricket Board, said: "I'm sure I speak for the entire cricket family when I say how sad I am to hear of the passing of the Duke of Edinburgh.
"His passion for the game we all love was well known and the trophies presented to the men's and women's county champions are a tribute to his dedication to our sport. We owe him a great debt for his support and passion over many decades."