The couple also reportedly want to choose an "ordinary" name for their baby.
Eugenie's baby will not automatically be entitled to a HRH title, as those only extend to grandchildren of the Queen. But even if the Queen offered their baby a title as a gift, it was "not Eugenie or Jack's desire" for their baby to have one.
Although they have royal titles, Eugenie and her sister Beatrice aren't currently considered senior working royals as they don't carry out duties on behalf of the Queen full time.
Ninth and tenth in line to the throne, both work fulltime and don't receive money from the government-supported fund that finances the Queen's official duties.
Eugenie is a director at Mayfair gallery Hauser & Wirth, having graduated with a degree in English literature and art history from Newcastle University in 2009.
She and Beatrice attend a few royal events each year, occasionally joining the Queen at a garden party, or making an appearance at Trooping the Colour with the other royals.
Jack is the European brand manager for Casamigos Tequila, co-founded by George Clooney.
Their child will be 11th in the order of succession - the first grandchild for Prince Andrew and Sarah Ferguson and the Queen's ninth great-grandchild.
Eugenie shared the news on Instagram last week, writing: "Jack and I are so excited for early 2021," along with photos of baby slippers and she and Jack smiling.
Her mum Fergie also shared her excitement, writing: "I am so excited by the news that Eugenie and Jack are expecting their first child.
"Thrilled for them both and in my 60th year cannot wait to be a grandmother. Welcoming a new baby into the York family is going to be a moment of profound joy."