The team is seeking a second-round pick in this year's draft as well as a pick in 2014 in any trade for its franchise player, league sources told ESPN NFL Insider Adam Schefter on Friday.
The Chiefs have had trade discussions with the Miami Dolphins, among other NFL teams, sources said. If the Chiefs could get back two picks, it would help make up for the second-round pick and conditional third-round pick that they gave up for quarterback Alex Smith.
Chiefs officials are at Texas A&M Friday to work out offensive tackle Luke Joeckel and plan to individually work out other premier left tackles.
Albert signed his franchise tender Thursday and told USA Today Sports that he's not "disgruntled," as he has been portrayed in media reports.
He signed his tender two weeks after the Chiefs placed the franchise tag on him. Albert will make $9.828 million if he plays under the one-year contract.
That is no sure thing, though. The Chiefs have spoken to his representatives about a long-term contract in addition to word that he may be traded.
"I want to know where I'm going, when I'm going, and I want some security," he told USA Today.
Albert is a former first-round pick who has locked down the left side of the Chiefs' offensive line the past five seasons. He played all 16 games two years ago but missed three games last season while dealing with back injuries.
Information from The Associated Press was used in this report.