* Graphic: World FX rates in 2018 tmsnrt.rs/2egbfVh
* Graphic: Trade-weighted sterling since Brexit vote tmsnrt.rs/2hwV9Hv
LONDON, Oct 1 (Reuters) - Sterling rose on Monday on reports Britain is preparing to make a significant offer to the EU to reach a Brexit deal but investors said a divisive ruling Conservative Party conference this week could weaken the currency.
Bloomberg reported that the UK government was preparing to back down on customs checks between mainland Britain and Northern Ireland, a major obstacle to Brexit that diplomats are trying to overcome before a European Union summit in October.
The pound jumped as much as 0.7 percent on the report to $1.3118 before relinquishing most of its gains to trade flat at $1.3030 at GMT 1545.
Against the euro it traded up 0.3 percent at 88.76 pence .
Brexit minister Dominic Raab said Britain would look at how regulatory checks could be used as part of a solution to the Irish border problem and accused the European Union of “weaponising” the issue.
At her party’s annual conference this week, May will defend her plan for leaving the EU. Critics like ex-foreign secretary Boris Johnson could openly defy her.
Six months before Britain is due to leave, currency traders are focused on events like the conference and an upcoming EU summit for signs of whether the UK will clinch a deal.
“Deep divisions within the party could turn this conference into a disaster for May, which could easily have negative effects for sterling,” said Thu Lan Nguyen, an FX strategist at Commerzbank AG in Frankfurt.
“The government might end up without any Brexit plan whatsoever. Whichever way one looks at it, the next few days will be uncomfortable for sterling.”
Sterling came under pressure last week as optimism for a Brexit trade deal faded and the dollar surged.
Other analysts said investors would refrain from trading on negative Brexit headlines that did not alter the odds of a no- deal Brexit.
“We would see any dips in the pound in and around the Tory Conference as an opportunity to buy the currency – with a deal on the Irish border backstop to see a sharp re-pricing to $1.36-1.38, “ said ING currency strategist Viraj Patel.
Finance minister Phillip Hammond told the BBC that doubts about Britain’s future had already hit the economy and the next few weeks would be crucial in securing an exit agreement.
“The mood is undoubtedly that people want to do a deal with the UK,” he said.
Businesses have expressed frustration over the government’s lack of clarity on Brexit and what some see as a decision to put ideology before the economy.
Last week, Britain’s opposition Labour Party sought to woo business bosses, although it also proposed a raft of radical economic policies, saying the Conservatives had given them the opportunity to pitch an alternative leftist strategy . (Reporting by Tom Finn; editing by Andrew Roche)