Cities have been falling over themselves to lure Jim Cramer's Action Alerts PLUS Charitable Trust Portfolio holding Amazon Inc.'s (AMZN) - Get Report proposed second headquarters to their town. But a new report from Zillow may dampen some enthusiasm local residents may have had for the move. 

Among the 20 cities that have been identified as finalists, Nashville, Denver and Los Angeles residents would see the highest increase in annual rent appreciation. 

"While the prospect of 50,000 new jobs is no trivial matter for any city, the home to Amazon's second headquarters likely won't experience the kind of dramatic boost in rents Amazon helped fuel in Seattle," said Zillow senior economist Aaron Terrazas. "Nashville and Denver are the only two metros that could see a greater boost to rents due to Amazon than Seattle has seen."

In Denver, rents could rise an additional 2.3 percentage points each year if Amazon selects the area. Los Angeles rents would be nearly 2% annually. 

Meanwhile, Nashville, which already has one of the fastest-appreciating housing markets in the country, could see the monthly median rent increase to $1,547 from $1,511. 

"Growth often brings growing pains, and in the past it has been lower-income households that bore the brunt of rising housing costs in the face of rapid expansion. Whichever community is chosen, it's critical that local leaders begin working now to prepare their cities as best they can," Terrazas said. 

The three cities expected to see the smallest median rent increases are Indianapolis, which would see rents increase to flat growth after a projected 0.2% decline without the company; Chicago, which is expected to see a 0.1% increase with Amazon vs. an expected 0.2% decrease without the company; and Toronto, which would see a 0.2% increase vs. an expected 1.4% increase without the company. 

"Many of the finalist cities have a relatively strong record of adding new rental supply in response to new demand, helping curb rapid rent growth - though rents could escalate more quickly in communities with a smaller employment base and/or those unable or unwilling to add meaningful new housing supply," Terrazas wrote.