The mother and her young daughter were a part of a family of four who fell ill while on holiday in the Red Sea resort of Hurghada, according to the Daily Telegraph.
They were among the guests evacuated from the Steigenberger Aqua Magic hotel after Burnley couple John and Susan Cooper died suddenly on August 21, and were told by an environmental health officer that samples showed they were suffering from the infection, the paper added.
Shigella is a highly infectious condition which can cause diarrhoea and stomach cramps, and is a common cause of food poisoning.
A lawyer for the unnamed family said it was a "crucial development", and could indicate that "pathogens were present at the property".
Nick Harris, from law firm Simpson Millar, told the paper: "If you have an illness problem in an all-inclusive property with several hundred guests moving around, you can either close the place for a deep clean or attempt to deal with it while the guests remain in situ.
"If you believe it's in the water, additional chlorine might be added to it in an attempt to kill the bug, so it's important to find out things such as what the Coopers drank that evening before they collapsed.
"If there was a sickness bug that the hotel knew about, how did they deal with it?"
Thomas Cook moved 300 guests out of the hotel as a precaution 24 hours after Mr and Mrs Cooper died after becoming aware of an increased number of illnesses.
Chief executive Peter Fankhauser previously confirmed that 13 customers had food poisoning but were not in a serious condition.
Mr Fankhauser flew to Cairo on Wednesday to discuss the deaths with prime minister Dr Mostafa Madbouly and minister of tourism Rania Al-Mashat.
Following the meeting, Ms Al-Mashat said "detailed autopsies" were being conducted by a team of forensic pathologists. The process is expected to be concluded next week.
She said: "When the pathologists have completed their detailed forensic analysis our priority will be, of course, to then contact the Cooper family in England to explain the findings as they, more than anyone, need to know what took away John and Susan.
"Their bodies will then be repatriated next week with the Cooper family in England."
A separate investigation led by Egyptian prosecutor Nabil Sadeq is testing food, water and air conditioning at the hotel.
This will be "robust, thorough and independent", Ms Al-Mashat insisted.
Mr Sadeq has previously said an inspection of the Coopers' hotel bedroom found no harmful gas emissions or leaks.
Ms Al-Mashat added: "It is crucial for everyone involved in the tragic passing away of John and Susan, none more so than the grieving Cooper family, that we get to the bottom of the matter and determine the truth based on evidence."
A Thomas Cook spokesman said Mr Fankhauser "reiterated his personal commitment, and the commitment of everyone at Thomas Cook, to get to the bottom of what went wrong".
The chief executive also met with British Ambassador to Egypt, John Casson, and Deputy Head of Mission, Helen Winterton.
Thomas Cook pledged to continue to work with the Egyptian authorities and the Foreign and Commonwealth Office to "prioritise the very best interests of the Cooper family".
The spokesman added: "The well-being of our customers in Egypt remains of paramount importance."
Thomas Cook has commissioned its own tests into food hygiene and air conditioning at the hotel, although it has not been granted access to the Coopers' room. The results are due in the middle of next week.