O'Ward was to have raced full-time in IndyCar this year with Harding Steinbrenner Racing before being released due to a funding shortfall, and his subsequent relationship with Red Bull was brief due to not being awarded sufficient Formula 1 superlicence points for his Indy Lights result.
Between the 20-year-old O'Ward and 23-year-old Askew, the driver line-up is set to be the youngest in IndyCar next season.
“As we look to establish Arrow McLaren SP and re-establish McLaren in the world of IndyCar, I'm delighted to be welcoming these two young, homegrown talents to the team,” said McLaren Racing's Sporting Director, Gil de Ferran.
“Oliver and Pato represent the new generation of IndyCar drivers. Proven winners and exciting prospects, I have no doubt that they will form an excellent pairing as we take on the 2020 season.
“Both drivers were super-impressive in Indy Lights, racking up 16 wins between them over the last two seasons. We have taken time and care to make a driver selection that we believe is in the best long-term interests of Arrow McLaren SP.”
The team, a partnership between McLaren Racing and IndyCar incumbents Schmidt Peterson Motorsports, will run two Chevrolet-powered cars.
Honda ambassador and SPM driver for the past five seasons, James Hinchcliffe, remains under contract but is free to pursue other seats.
“I have the utmost respect for James and would like to thank him for his hard work and accomplishments during that period and wish him well in his future endeavours,” said co-owner Sam Schmidt.
“We recognise that James is a fiercely motivated and determined competitor, and we won't hesitate to release him unconditionally to secure another drive, whether in IndyCar or another series.”