• CrowdStrike's attorney said Delta turned down on-site assistance in the wake of the outage.
• Delta CEO Ed Bastian estimated last week that the mass cancellations after the outage cost the company some $500 million, and CrowdStrike's lawyer says its liability is capped at under $10 million.
• Delta canceled more than 5,000 flights after the outage, taking longer to recover than rivals.

CrowdStrike on Sunday said Delta Air Lines had rejected on-site help during last month's massive outage that sparked thousands of flight cancellations.

Delta CEO Ed Bastian told CNBC's "Squawk Box" last week that the mass cancellations following the outage, which occurred at one of the busiest times of the year, cost the company about $500 million, including customer compensation. The airline has "no choice" but to seek damages, he said.

Bastian told staff on Friday that the airline had informed CrowdStrike and Microsoft that the company was "planning to pursue legal claims" to recover its losses stemming from the outage and that it had hired law firm Boies Schiller Flexner.

In response, Michael Carlinsky, CrowdStrike's lawyer and co-managing partner at Quinn Emanuel Urquhart & Sullivan, wrote to Delta's lawyer David Boies on Sunday that Delta's litigation threats "contributed to a misleading narrative that CrowdStrike is responsible for Delta's IT decisions and response to the outage."

He said CrowdStrike CEO George Kurtz reached out to Bastian to "offer onsite assistance, but received no response."

Delta canceled more than 5,000 flights between the July 19 outage, caused by a botched software update, through July 25, more than its rivals.

CrowdStrike shares have lost more than 36% of their value since the outages affected millions of computers running the company's software atop Microsoft's Windows operating system. The outage hit industries from banking to health care to air travel.

"Should Delta pursue this path, Delta will have to explain to the public, its shareholders, and ultimately a jury why CrowdStrike took responsibility for its actions—swiftly, transparently, and constructively—while Delta did not," Carlinsky's letter said.

He said Delta would have to preserve a series of documents, including those describing its information-technology infrastructure, IT business continuity plans and its handling of outages over the past five years.

CrowdStrike's contractual liability is capped in the single-digit millions, the letter said. Delta did not comment on the letter on Sunday night. In a separate statement, CrowdStrike said it hopes "Delta will agree to work cooperatively to find a resolution."

"We did everything we could to take care of our customers over that time frame," Bastian said in an interview Wednesday on CNBC's "Squawk Box." "If you're going to be having access, priority access, to the Delta ecosystem in terms of technology, you've got to test this stuff. You can't come into a mission critical 24/7 operation and tell us we have a bug. It doesn't work."

CrowdStrike vowed to release future software updates in stages in a preliminary post-incident report.

On July 30, CrowdStrike shareholders filed a suit against the company in a Texas federal court and sought damages for declines in their investments.

CrowdStrike reports fiscal second-quarter results Aug. 28.

A Microsoft spokesperson did not immediately respond to CNBC's request for comment.