A new report from real estate data firm CoreLogic shows about 2.1 percent of all Massachusetts homes with a mortgage, or about 7,851 homes, were in the foreclosure inventory as of September, lower than the 3.3 percent national average.

Nationwide, the foreclosure inventory has been declining. In total, about 1.4 million homes were in the national foreclosure inventory in September, compared with 1.5 million, or 3.5 percent in September 2011. Month over month, the national foreclosure inventory was down 1.1 percent from August. The foreclosure inventory is the share of all mortgaged homes in any stage of the foreclosure process.

In Massachusetts, the foreclosure inventory trend is heading upward slightly, with a .2 percent increase in the foreclosure inventory from a year ago.  

"The continuing downward trend in foreclosures, along with a gradual clearing of the shadow inventory are signs of stabilization and improvement in the housing market," Anand Nallathambi, president and CEO of CoreLogic, said in a statement. "Increasingly improving market conditions and industry and government policy are allowing distressed homeowners to pursue refinancing, loan modifications or short sales rather than foreclosures."

"Homes lost to foreclosure in September 2012 are down 50 percent since the peak month in September 2010, and 22 percent less than the beginning of the year," said Mark Fleming, chief economist for CoreLogic.

The five states with the highest number of completed foreclosures for the 12 months ending in September were: California (108,000), Florida (92,000), Texas (59,000), Georgia (55,000) and Michigan (51,000). These five states account for 47.7 percent of all completed foreclosures nationally.