Sancho, 20, is expected to leave Dortmund this summer in a deal worth more than £100 million and Chelsea have been monitoring his situation for months with a view to making an offer.
However, sources have told ESPN that senior figures at Stamford Bridge believe United are winning the race to sign Sancho, and while they have not entirely given up hope, other alternatives are now being discussed.
The Bundesliga side are unwilling to meet a £97m buyout clause they agreed for Coutinho, who is already planning for life away from the club with his agent Kia Joorabchian sounding out possible suitors.
Coutinho left Liverpool for Barcelona three years ago for £145m but is open to joining another Premier League team. Bayern paid Barcelona an £8m loan fee and wages in excess of £200,000-a-week, neither of which would be an obstacle for Chelsea if a similar deal could be reached.
Barcelona would prefer to sell, however, although it is unclear at this stage what their asking price would be. They are keen to radically overhaul their squad and could be willing to negotiate to help fund moves for fresh players.
Chelsea are financially secure themselves, but there are wider concerns about the impact the coronavirus pandemic could have on football.
The Blues recorded a loss of £96.6m in their latest financial accounts, for the year ending 30 June 2019, as the impact of missing out on Champions League football was keenly felt.
Since then, the club has sold Eden Hazard to Real Madrid for a fee which could rise to £150m and has chosen not to sign anyone in January despite being able to do so after reducing their FIFA transfer ban on appeal at the Court of Arbitration for Sport.
They were therefore not opposed to sanctioning a £100m move for Sancho, but the spread of the coronavirus has complicated the picture.
The suspension of football across Europe combined with the probability of a return to action behind closed doors has led to fears the transfer market may stagnate and triggered fresh concerns over the sensibility of committing such big money on one player.
Chelsea have moved to a more financially prudent business model in recent years rather than relying on the personal wealth of owner Roman Abramovich and are in discussions with their playing staff over a wage cut or deferral due to the pandemic.
Sancho was a boyhood Chelsea fan, but United's greater financial might may prove decisive; Chelsea are one of the biggest clubs in the world but their turnover was £446.7m compared to United's £627m for the same period.