The Edinburgh side began the season with a 10-point penalty for going into administration, with their financial problems leading to the creation of the Foundation of Hearts.
More than 7,000 supporters have already signed up to the foundation, which was named preferred bidder for Hearts earlier in August.
Southern is one of those and he believes the club can be run successfully for "many generations to come" if enough supporters pledge their support.
"The departure of the former owner (Vladimir Romanov) and board presents an opportunity to establish certainty and stabilise the club for future generations," he said in an open letter to fans.
"We are now being asked to step forward when our club needs us most. There is a 'cost' but, in my own view, it is negligible when compared to the cost of not securing the club now.
"This is our moment. If we hesitate, it may not just be the moment that is gone.
"Hearts is a club of the people, it always has been. If anyone ever needed proof of that it has been demonstrated time and time again particularly in recent months, and not simply in financial terms.
"As Hearts supporters you can rightly take pride in how you do things, not just what you do.
"I know I speak on behalf of the staff at Tynecastle when I thank you for your ongoing efforts to secure a more stable future for the club.
"The dignity shown throughout such trying times is as valuable as any sum of money that has been raised.
"For this reason, it is only right that you are now in pole position to take ownership of the club.
"There is no reason why Hearts should not remain as one of the biggest and potentially most successful clubs in Scottish football."