The former is struggling to stay relevant in an era of online shopping and low-cost fashion. The latter excels in the digital forum, but faces limitations in terms of its physical footprint.
To solve these issues, retail expert Robin Lewis offered what he called a "win-win" solution: Amazon should acquire Sears.
"The acquisition becomes [Amazon CEO Jeff] Bezos' answer to omnichannel and the proven revenue synergy of consumers' ability to shop online and off; the convenience of proximity for pick up and returns; and facilitation of even greater delivery speed," Lewis wrote in his online newsletter, The Robin Report. "So just as Walmart's 4,500 stores double as distribution centers, so would Amazon's acquired Sears/Kmart stores."
Sears boasts more than 2,000 stores in its retail footprint, when accounting for both its Sears and Kmart nameplates. Amazon, on the other hand, only operates domestic fulfillment centers in 14 states.
"What Eddie [Lampert, Sears CEO] gets in such a sale is a potentially profitable exit strategy that many analysts, myself included, believe he is pursuing," Lewis said. "This could fall nicely into Bezos' hungry little hands. Amazon might be able to cut an incredible deal, at least far less costly in time and capital, than building or leasing its own nationwide distribution centers/stores."
A Sears spokesperson said the company does not comment on speculation and rumors. Amazon did not immediately respond to a request for comment.
Sears shares rose about 12 percent to $37.60 on Thursday, after news broke that director Thomas Tisch bought 475,000 shares of the company.
Andrew Wilkinson, chief market analyst at Interactive Brokers, noted that not only are the shares moving, but there is some notable options activity as well, with indications of higher implied volatility, a key driver of options premiums. This could imply traders are betting on a change of ownership, he said.
"There is a jump in implied volatility moving in the same direction as the underlying share price. There is call-option activity that was non-existent prior to today's share price move," he added.
"I would say options are pricing quite defensively here, so we've got sizeable positions, or action in May, or at the money," Wilkinson said. "Not only are shares rallying, but options are pointing to a greater chance of a sharp move up or down in the very near term."
According to Wilkinson, 16,000 contracts traded for Sears, most notably the May 16 expiration.
"A potential bid would [substantiate] that," Wilkinson said.
Lewis said that if an acquisition were to happen, the Sears and Kmart nameplates should be done away with, as Amazon's brand power is much stronger. Retail Metrics president, Ken Perkins, agreed that if a deal were done in such a way, it could have some merit.
"I've got to believe that Sears would be open to all options at this point," Perkins said. "It really would depend on what kind of price Amazon could offer, and would it be worth their while to do that."
Sears posted a loss of $930 million for the year ended Feb. 1.
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If a deal were to happen, it could be a further blow to bricks-and-mortar players, which have been undercut by Amazon. Many analysts have cited physical stores' expansive retail footprint as one of their few advantages in competing with the online giant.
"Depending on how many physical locations they took over it certainly would give [Amazon] another leg up," Perkins said. "It certainly would ramp up the pressure on the rest of the retailers that are trying to build out their omnichannel businesses."
He added, however, that a deal would also ramp up Amazon's costs. He also pointed out that the Sears footprint is so large, it would leave Amazon with a lot of square footage to fill.
Forrester analyst Sucharita Mulpuru said she doesn't see a deal between the two happening, adding that Amazon acquires more entrepreneurial companies.
"Sears continues to try to think it has a future as a retailer but its real estate is probably better deployed in another way," Mulpuru said.
Belus Capital Advisors analyst Brian Sozzi sounded a similar note.
"James Cash Penney has a better chance of rising from the grave and personally delivering a check to Edward Lampert to buy all of Sears than Amazon offering up even a bid for the company," he said. "Retrofitting these stores into Amazon fulfillment operations all at once so as to start driving value, is a task and extra financial burden that Amazon doesn't need or want."
Lewis has worked in the retail industry for more than 40 years, including consulting experience. He has served in executive positions at VF Corp, and serves as a professor at the Fashion Institute of Technology.
—By CNBC's Krystina Gustafson. CNBC's Kate Gibson contributed to this report.