The former England international's wages are thought to have been a stumbling block but the left-back also preferred to remain with the Foxes rather than join their Sky Bet Championship rivals.
QPR have been linked with Konchesky recently but it has emerged that the Seagulls were in contact with Leicester last week regarding the 32-year-old's availability.
It is widely accepted that the Foxes have had to trim their wage bill in order to fall in line with the new Financial Fair Play regulations.
Jermaine Beckford and Richie Wellens left for precisely that reason over the summer - striker Beckford sold to Bolton and central midfielder Wellens having the final year of his contract settled.
Konchesky is also a big earner and it has been speculated that he could be the next to depart the King Power Stadium.
He was left out of the squad that travelled to Derby on Saturday, having started the first two league matches for Nigel Pearson's side.
Leicester boss Pearson revealed after his team's 1-0 victory at Pride Park that Konchesky was omitted due to interest from another club which could result in the player leaving.
He said: "There has been speculation about potential moves away and I thought it was in the best interests of the player and ourselves that he was omitted from the squad.
"I can confirm that is the case (that he was omitted because of a potential move), but he is still our player and I will speak to Paul again on Monday."
Konchesky joined Leicester from Liverpool for a reported fee of £1.5million in July 2011. He was signed by former Foxes boss Sven-Goran Eriksson.