That is according to BBC reporter David Ornstein.
Arsenal finished fifth in the Premier League last season, missing out on a spot in the top four after defeats to Wolves, Crystal Palace and Leicester at the business end of the season.
As a direct consequence, the north London outfit will not be able to enjoy the same type of extravagant transfer budget as other teams at the summit of England’s top flight.
This means a move for Zaha looks to be out of the question, as Palace hold firm on their £80m valuation of the winger.
The Eagles are also unlikely to want their star man to leave this summer due to the impending departure of Aaron Wan-Bissaka to Manchester United.
But Arsenal chiefs feel they could have landed Zaha had they finished in the top four last month.
“Zaha is keen to leave Selhurst Park this summer and favours a move to Arsenal, who in turn would like to recruit him,” Ornstein said.
“However, Palace value the Ivory Coast international at up to £80m and – in light of Wan-Bissaka’s departure – are reluctant to entertain offers this summer.
“It is understood that Arsenal were optimistic they would have the necessary resources to bring Zaha in if they had qualified for next season’s Champions League.
“Yet a failure to do so has restricted their budget to around £45m, meaning they would need to raise substantial funds through players sales to even tempt Palace into doing business for the 26-year-old.”
Pierre-Emerick Aubameyang has been linked with a shock Arsenal exit this summer, while Gunners team-mate Lucas Torreira is reportedly unhappy with life in London.
The former continues to be eyed by Chinese clubs.
Torreira, meanwhile, looks set to return to Serie A by landing at AC Milan.
And Ornstein thinks both players could be used to bring in much-needed funds for Emery.
“The most realistic way of generating the required finance would be by cashing in on Pierre-Emerick Aubameyang, though the implications of parting with last season’s top scorer would present a major dilemma for the Gunners,” he added.
“Another of their assets is Uruguay midfielder Lucas Torreira who arrived at the Emirates Stadium from Sampdoria last summer and has been linked with a return to Italy.”
Arsenal continue to be linked with other wingers, in the form of Yannick Carrasco, Ryan Fraser and Ivan Perisic.