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According to The Sun, the 30-year-old actor, musician and LGBTQ+ advocate is close to landing the leading role after pay negotiations with BBC bosses reached an 'advanced stage'.
If Olly does take on the character, he'll be the second star to make Doctor Who history, following on from Jodie Whittaker who was announced as the first ever female Time Lord in 2017
After appearing in 23 episodes of the long-running BBC sci-fi drama, earlier this year it was suggested that Jodie's time on the show was coming to an end.
Reports state that the 39-year-old will stand down after two specials in 2022, leaving the spot wide open for Olly to fill.
While we've yet to receive confirmation Alexander is cast as the 14th Doctor, a source told The Sun: "Olly is on the verge of being announced as the new Doctor. Negotiations are at an advanced stage.
"He's been heavily promoted for the role by Russell - who he's worked closely with on other projects. Russell was impressed with his work and felt he was a great fit for the Doctor.
"It's definitely happening, he's just had to keep quiet about it while negotiations were finalised."
The star first found fame as the frontman for electro band Years and Years, and rose to critical acclaim earlier this year for his performance in Russell T Davies' Channel 4 drama It's a Sin.
Olly starred as the lead Ritchie Tozer, one of five 18-year-olds who move to London in 1981.
Dubbing their home 'the Pink Palace' and hosting regular parties, their lives are turned upside down by the AIDS crisis, with the drama exploring the impact the virus had on the LGBTQ+ community in that time.
Russell, a former head writer and executive producer on Doctor Who, saw the potential in Olly and supported the idea of him taking over as the Doctor.
He told SFX magazine earlier this year: "Olly would make a marvellous Doctor Who."
Quite frankly, fans are delighted at the idea of Olly being the new Time Lord, with one writing on Twitter: "Olly Alexander is the new #DoctorWho AND I LOVE IT!"
Another wrote: "I'm acc gonna be so happy if Olly Alexander is the new Doctor."
A BBC insider said: "As ever bosses are staying tight lipped about what they have planned but with filming still going on they clearly have plans for episodes to be playing out much later into 2022 so there's lots more to come for Jodie's Doctor."
The BBC declined to comment when approached by LADbible.