With regard to Mexican California in the 1840s, Hague and Langum's Thomas O. Larkin says, on page 10:
The United States had long been interested in California and had more than once attempted to purchase it.
I did some searching and found nothing more on the topic. Who offered what to whom?
The short answer is 'Yes'.
The initiative was taken by James K. Polk who sent an envoy, John Slidell, in November 1845 following Mexico's severing of diplomatic relations with the US. Referring to Texas,
Polk had instructed Slidell to insist that the Mexicans recognize annexation as a fait accompli. Most surprisingly, Slidell had also been told to purchase Upper California and New Mexico and to use the perennial complaint of unpaid American claims against Mexico to pressure and facilitate the sale.
Source: Brian Delay, 'War of a Thousand Deserts: Indian Raids and the U. S. -Mexican War' (2008)
Citing B. Mills in U.S.-Mexican War (2003), Wikipedia states that Polk authorized Slidell
to offer Mexico around $5 million for the territory of Nuevo México and up to $40 million for Alta California.
Although Polk had insisted on secrecy, the press found out before Slidell had even arrived in Veracruz. This caused a storm in Mexico where the administration of President Herera was already pressure due to its failed policy towards Texas. Consequently, the Mexican government
refused to receive Slidell in his formal capacity, but the mere suggestion of selling national territory undermined the tottering administration all the more.
Source: Delay
Then,
In January 1846, shortly after the Mexican president José Joaquín de Herrera had refused to receive Slidell in his official capacity, but before the envoy finally abandoned his mission, Polk ordered General Zachary Taylor to march his forces to the Rio Grande. This extraordinarily provocative move made war all but inevitable.
Source: Delay
A less direct attempt had been made to buy California in 1842. This came about due to numerous incidents of stolen American property in Mexico and American ships being seized in Mexican ports. As a result,
In 1839 there was arbitration between the two countries, and an award of $2 million to the Americans, to be paid in 20 installments. Mexico quickly fell behind on the payments, and new claims continued to pile up. In 1842, a frustrated President Tyler proposed that Mexico hand over California as compensation, with the US government to pay off American claims, but the Mexicans refused.
Yes. In 1845, the U.S. offered to buy (Alta) California for $25 million. (This followed a $5 million offer in 1835.) That would have been part of the U.S. modus operandi and a sensible thing to do, because the Mexican War cost $100 million, plus American (and Mexican) lives. But having (barely) acceded to the loss of "Texas," Mexico wasn't about to give up "California" without a fight.
The treaty of Guadaloupe-Hidalgo was not characterized as an annexation but a (forced) sale of the conquered territory for $15 million, following the Louisiana Purchase for the same nominal amount. Later, in 1853, the Americans made the Gadsden Purchase to straighten out the boundaries and a potential railroad line through New Mexico for another $10 million.
