The two-time NL Cy Young Award winner's request neared the record amount sought in arbitration. Houston pitcher Roger Clemens asked for $22 million in 2005.
San Francisco's offer was the highest in arbitration history, topping the $14.25 million the New York Yankees proposed for shortstop Derek Jeter in 2001.
"I'm overall optimistic that we'll find common ground without a hearing room," Bobby Evans, Giants vice-president of baseball operations, said before seeing Lincecum's filing numbers. "It's a process that begins long before today in terms of conversations about possible deals that work for both sides. That process has continued in a mutual fashion. At this point we haven't reached a conclusion."
Also Tuesday, the Giants and slugger Pablo Sandoval agreed on a three-year contract. The 25-year-old third baseman became an All-Star last season after losing nearly 40 pounds during a rigorous off-season regimen. He batted .315 with 23 home runs and 70 RBIs in 2011.
Lincecum, the winning pitcher in the Game 5 World Series clincher at Texas in 2010, earned $13.1 million last season when he completed a two-year deal worth $23.2 million.
San Francisco's front office would like to lock up the 27-year-old Lincecum and fellow starter Matt Cain with long-term deals. Lincecum seems set on keeping his options open in the near future on a shorter contract.
"We know we'll at least have a one-year deal," Evans said. "I can't really predict where it will end up. In this process your two parties are always filing to try to come to a midpoint. The negotiation is really about the midpoint."
With Lincecum earning a hefty contract, Evans joked, "I usually leave off the final three zeroes because it's easier to calculate."
If the past is any indication, the sides will do their best to reach agreement before spring training and before an arbitration hearing.
In February 2010, Lincecum agreed to a $23 million, two-year contract ahead of the scheduled hearing. He had been set at that time to ask for $13 million.
That last contract was quite a raise for the undersized, hard-throwing pitcher his teammates call "Franchise" and "Freak" after he earned $650,000 in 2009.
"We're looking at different player contracts that give us an idea where we think Tim should be," Evans said. "There is not ever a player that's exactly like the one you have. Ultimately there is only one guy that looks just like him."
Lincecum — the 10th overall draft pick out of Washington in 2006 — has been an All-Star in each of the past four seasons. He went 13-14 with a 2.74 ERA last year for his first losing record. The Giants scored no runs while he was in the game in seven of 33 starts, had one run six times and two runs five times, according to STATS LLC.
Also Tuesday, the Giants reached one-year agreements to avoid arbitration with outfielders Melky Cabrera and Nate Schierholtz and reliever Santiago Casilla.
Cabrera agreed to a $6 million deal.
San Francisco, which sold out every game in 2011 but missed the playoffs, will have a payroll of around $130 million.
"Obviously the revenue that has been generated by our ownership and the support of our fans here makes the payroll level we have possible," Evans said. "We don't take that for granted. We know that with that kind of payroll comes responsibility and expectation."